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Elasticity and Equation of State

Elasticity and Equation of State. Lars Stixrude University of Michigan. Probe: Earthquakes. www.iris.edu. Detector: Seismograph. Determine: Elastic Wave Velocities. To a good approximation: Radially homnogeneous Isotropic Small but important: Lateral heterogeneity Anisotropy. F = kx.

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Elasticity and Equation of State

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  1. Elasticity and Equation of State Lars Stixrude University of Michigan

  2. Probe: Earthquakes www.iris.edu

  3. Detector: Seismograph

  4. Determine: Elastic Wave Velocities To a good approximation: Radially homnogeneous Isotropic Small but important: Lateral heterogeneity Anisotropy

  5. F = kx Elastic Constants Elastic Constant Tensor Soft Stress Tensor Strain Tensor Stiff

  6. Uniaxial stress (P wave) • 11=c1111 11 • 11=c1122 22 • Poisson’s Ratio • =-22/11 = c1122/c1111 • Fluid: =0.5 • i.e. volume conservation • 22=33=-11/2 • Crystals don’t conserve volume on uniaxial strain • ~ 0.25 Crystals stiffer, faster P-wave velocity Elasticity x,1 11 y,2 22

  7. Shear stress (S-wave) 12=c1212 12 Produces no other strains for crystals with symmetry higher than monoclinic Fluid No restoring force for shear stress Vanishing S-wave velocity Elasticity x,1 12 y,2

  8. Symmetry of elastic constant tensor cijkl Unchanged with respect to interchange of: i,j; k,l; ij,kl Crystalline symmetry: maximum 21 independent elastic constants Cubic: 1,2,3 directions equivalent Three independent elastic constants c1111 (=c2222=c3333), c1122, c1212 Isotropic material: two independent elastic constants Bulk modulus: K=(c1111+2c1122)/3 Shear modulus: G=(c1111-c1122)/2=c1212

  9. Upper Mantle Xenolith, Depth ~ 100 kmRed=garnet (gt); black=orthopyroxene (opx); green=clinopyroxene (cpx); yellow-green=olivine (ol)

  10. Isotropic Aggregates • Observation: Earth is nearly isotropic • But crystals are not! • They must be nearly randomly oriented • How to relate crystal cijkl to isotropic K,G? • How to “average” over elastic properties of coexisting crystals? • Constant stress (Voigt) • Constant strain (Reuss)

  11. Influence of pressure • How to represent this data with a simple functional form? • Straight lines? Circles: Karki et al., 1997, Am. Min. Squares: Murakami et al., 2006, in review

  12. Influence of pressure • How to represent this data with a simple functional form? • Straight lines? • Back up, remembering self-consistency Circles: Karki et al., 1997, Am. Min. Squares: Murakami et al., 2006, in review

  13. Influence of pressure • Eulerian finite strain formulation works for elasticity as well • Parameters: Moduli and first pressure derivatives. • Self-consistent expressions essential Stixrude & Lithgow-Bertelloni, 2005, GJI Circles: Karki et al., 1997, Am. Min. Squares: Murakami et al., 2006, in review

  14. Influence of temperature • Most important effect: • Change in volume (density) • Moduli co-vary with density at similar rates whether density is altered by pressure or temperature • Microscopic picture: Lower density means weaker bonds and smaller moduli

  15. Influence of composition • Iron has by far the largest influence • Most massive major element • Large influence on density • Increasing Fe content decreases velocities • Fe also lowers shear modulus

  16. Influence of crystal structure • For crystals of same mean atomic weight • Scaling with density • Birch’s law

  17. Lateral Heterogeneity Ritsema

  18. Lateral Heterogeneity

  19. Influence of Temperature • VS and VP should co-vary with temperature! (both depend on G) • But they don’t near core-mantle boundary • Lateral heterogeneity must have other origin • Composition (iron content, reaction with core?) • Phase (partial melt?)

  20. Ultra-Low Velocity Zone (ULVZ) • Very thin (< 40 km) • Located right at core-mantle boundary • Extremely low VP (-10 %) • VS may be >20 % lower than normal • Partial melt Williams and Garnero (1996) Science

  21. CaSiO3 Perovskite Transition Cause of lower mantle reflectors? Stixrude et al. (2007) Phys. Rev. B

  22. Acoustic Wave Velocities Polarization Directions wi Propagation Direction ni

  23. Anisotropy Polarization Directions wi Propagation Direction ni Azimuthal: Dependence on n Polarization: Dependence on w Only for shear: shear wave splitting

  24. Polarization Anisotropy • VSV<VSH • Explain by olivine • If olivine b axis aligned horizontally • Horizontally polarized S-waves faster than vertically polarized Ekstrom and Dziewonski (1999) Nature

  25. Anisotropy and Deformation of Olivine

  26. Olivine, Mg2SiO4 Fastest direction Compress Mg- and Si-polyhedra Easiest dislocation glide direction Shortest repeat distance

  27. Olivine at 11 GPa (~300 km depth) Easy slip along c! Fastest direction perpendicular to flow! Mainprice et al. (2005)

  28. Detection of Water? Wood (1995)

  29. New phases • Post-perovskite MgSiO3 • Transition near base of mantle • Layered, presumably strongly anisotropic • Possible implications for D’’ structure Pbnm Cmcm Murakami et al. (2004) Science

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