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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 Elasticity stress, strain, time

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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

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