1. Lectures on Rheology of Earth Materials Fundamentals and frontiers in the study of deformation of minerals and rocks (at Tohoku University) Shun-ichiro Karato Yale University Department of Geology & Geophysics New Haven, CT, U.S.A. June 25-27, 2003

2. Lecture Outline 1. Why rheology? 2. General background defects and plastic deformation thermodynamics 3. Some fundamentals of creep 3-1. Diffusion and diffusion creep 3-2. Dislocations, slip systems, and dislocation creep 3-3. Deformation mechanism maps 3-4. Effects of phase transformations 3-5. Effects of pressure, water

3. 4. Physical processes controlling the grain size 4-1. Grain-growth 4-2. Dynamic recrystallization 4-3. Nucleation-growth 5. Some applications 5-1. Lithosphere-asthenosphere 5-2. Rheology subducting slabs 5-3. Some unresolved problems

4. レオロジーでは時間依存性のある性質を扱うので 実験結果を直接地球には応用できない。 実験室での結果を地球に応用するときに物理的 モデルにもとづいたスケーリング則を確立しておく 必要がある。 例：流動則、結晶粒径（相転移） 要点 １。スケーリング則

5. 要点 ２。 レオロジー的性質は温度、水、相転移 、結晶粒径などをとおして地球の進化 、ダイナミクスと密接に関係している 。（密度や弾性的性質などと違う。）

6. Why rheology? Rheology controls mantle convection. –Mixing of geochemical reservoirs Rheology has strong influence on the way in which seismic wave propagation is affected by mantle convection. –anelasticity, anisotropy

7. What causes velocity heterogeneity: thermal or chemical origin (or both) ? Much of the velocity heterogeneity in the upper mantle has chemical origin (ocean vs. continent). To what extent is the deep mantle chemically heterogeneous? In regions where there is chemical heterogeneity, what is the cause of heterogeneity (which elements or which minerals)? Ritsema et al. (1999)

8. Models to explain geophysical and geochemical observations (Tackley, 2000)

10. Upper mantleTransition zone Trampert and van Heijst (2002) Pattern of anisotropy is very different between the upper mantle and the transition zone. What does it tell us?

11. What do we want to know? How does rheology control convection pattern? –What is the rheological stratification in Earth? –How can we infer flow pattern from seismological observations? How does rheology affect mixing? –Where are geochemical reservoirs and how have they survived (or not survived)? Why plate tectonics on Earth and not on Venus? –What are the mechanisms of localization of deformation? How have continents survived for billions of years? –Is continental lithosphere ”dry” or “wet”?

12. Frontiers in the study of deformation of minerals and rocks Extreme conditions –High-pressure(whole Earth dynamics) –Large strain(anisotropy, grain-size evolution, strain partitioning) –Small strain(seismic tomography-anelasticity post-glacial rebound) Complexities –deformation  chemical reaction(s) [metamorphism, phase transformations, partial melting, hydrogen redistribution---] –deformation  microstructural evolution [instability, localization: “plate” formation, deep earthquakes]

13. Some fundamentals of mineral and rock rheology T-P dependence of plastic deformation Stress, grain-size dependence Dependence on chemical environment Effects of phase transformations

14. Elastic deformation Plastic deformation: involves atomic jumps thermally activated processes probability T-dependence  time-dependence

15. Temperature dependence of elastic constant and viscosity: ductile strength (viscosity) is highly sensitive to T

16. Pressure dependence of elastic constant and viscosity: ductile strength (viscosity) is highly sensitive to P

17. Basic physics of plastic deformation Defects and plastic deformation –thermo-chemical equilibria –law of mass action –fugacity Thermally activated processes –origin of t-T dependence –Boltzmann statistics

20. Point defects

23. Effects of oxygen fugacity on defect concentration

24. The Kröger-Vink Diagram for olivine

25. Effects of water

27. A Kröger-Vink Diagram for a water bearing system

28. Diffusion creep

30. m=2 for diffusion inside of grains, m=3 for diffusion along the grain-boundary