Beyond Elasticity stress, 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 RheologyTomographyPhase Transitions Thermoelastic Convection Seismic Anisotropy Earth ’ s mantle and stress Anelasticity

Time scales IN EARTH Seismic waves 1 sec – 1000 sec. Earthquakes 10 sec – 1000 sec Plate tectonics 10 7 sec – sec IN LAB Acoustic velocity sec – sec Rock mechanics 1  sec – 1 msec Ductile flow 10 3 sec – 10 6 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=Vp 2 -4/3Vs 2

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

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?

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 Or Melting? Melts?

From Hirschmann, 2000

sp olol cpx opx

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

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

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

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 Circle Lattice spacings for stressed sample Stressed sample

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 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

Forced oscillation on MgO and Al 2 O 3 T= 800 o C P = 5GPa Frequency = mHz

Li Li et al 2009

MgO

Measure Amplitude of Diffraction Peaks with Time and Temperature 700 C

Measure Amplitude of Diffraction Peaks with Time and Temperature