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Elasticity: Equations of State in Geophysics

Learn about elasticity, equations of state, and seismic wave velocities. Explore how crystals affect material properties and study the influence of pressure, temperature, composition, and crystal structure. Uncover the mysteries of lateral heterogeneity and ultra-low velocity zones in the Earth's mantle.

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Elasticity: Equations of State in Geophysics

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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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