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Beyond Elasticity stress, strain, time. Don Weidner Stony Brook. From Don Anderson’s book ch. 14.
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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 Rheology Tomography Phase Transitions Thermoelastic Convection Seismic Anisotropy Anelasticity Earth’s mantle and stress Q, Vp,Vs
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
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)
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 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
To model Velocity • Phase diagram and Elasticity are not enough • Time scales of the phase transitions are also important
Melts? Is the low velocity zone due to Or Melting?
sp ol opx cpx
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)
Challenges for Experiments at deep Earth conditions of P and T Measure Deformation in situ Measure Stress Deform at a constant slow rate
Measurement of Stress s = F/A
Measurement of Stress s = M*e X-rays define d, lattice spacings, and can be used to define elastic strain.
Ideal Circle Stressed sample Lattice spacings for stressed sample
Challenges for Experiments at deep Earth conditions of P and T Measure Deformation in situ Measure Stress Deform at a constant slow rate
Multi SSD Press Sample gold foil Sample gold foil
Challenges for Experiments at deep Earth conditions of P and T Measure Deformation in situ Measure Stress Deform at a constant slow rate
Forced oscillation on MgO and Al2O3 T= 800 oC P = 5GPa Frequency = 10-100mHz
MgO Li Li et al 2009
Measure Amplitude of Diffraction Peaks with Time and Temperature 700 C
Measure Amplitude of Diffraction Peaks with Time and Temperature
Measure Amplitude of Diffraction Peaks with Time and Temperature