1. Beyond Elasticity stress, 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 Q, Vp,Vs RheologyTomographyPhase Transitions Thermoelastic Convection Seismic Anisotropy Earth ’ s mantle and stress Anelasticity

4. Time scales IN EARTH Seismic waves 1 sec – 1000 sec. Earthquakes 10 sec – 1000 sec Plate tectonics 10 7 sec – 10 16 sec IN LAB Acoustic velocity 10 -9 sec – 10 -6 sec Rock mechanics 1  sec – 1 msec Ductile flow 10 3 sec – 10 6 sec

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=Vp 2 -4/3Vs 2

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 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

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

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

14. From Hirschmann, 2000

18. sp olol cpx opx

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

23. Measurement of Stress  = F/A

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

25. Ideal Circle Lattice spacings for stressed sample Stressed sample

26. Conical Slits

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

29. Multi SSD Press Sample gold foil

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

36. Measurement of Stress by Proxy

37. Forced oscillation on MgO and Al 2 O 3 T= 800 o C P = 5GPa Frequency = 10-100mHz

38. Li Li et al 2009

39. MgO

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

41. Measure Amplitude of Diffraction Peaks with Time and Temperature