1. Emergent Anisotropy and Flow Alignment in Viscous Rock by Hans Mühlhaus, Louis Moresi, Miroslav Cada May 5-10

2. Outline The last time…. (Hakone): Folding instabilities in layered rock using director theory combined with pressure solution, mobile and immobile phases, novel computational scheme Publications: - Louis Moresi, Frédéric Dufour, Hans Mühlhaus, Mantle convection models with viscoelastic/brittle lithosphere: Numerical methodology and plate tectonic modeling, PAGEOPH, submitted 2001 - Muhlhaus,H-B, Dufour,F, Moresi, L, Hobbs, BE (2001) A director theory for viscoelastic folding instabilities in multilayered rock (30 pages) submitted to the Int. J. Solids and Structures -H-B Mühlhaus, L.N Moresi, B. Hobbs, and F. Dufour (2000)Large Amplitude Folding in Finely Layered Viscoelastic Rock Structures, PAGEOPH, submitted 2001 -Hobbs, B.E., Muhlhaus,H-B, Ord, A and Moresi, L. (2000) The Influence of Chemical migration upon Fold Evolution in Multi-layered Materials. Vol. 11, Yearbook of Self Organisation. Eds H.J. Krug and J.H. Kruhl; Duncker&Humblot, Berlin, 229-252 Today: Oriented materials and emergent anisotropy in simple shear and natural convection; thermal coupling in simple shear and …convection

3. Finite Anisotropy Director evolution n : the director of the anisotropy W, W n : spin and director spin D, D ’: stretching and its deviatoric part

4. Rotations Spin of an infinitesimal volume element average Spin of microstructure n n Undeformed ground state:

5. Anisotropic Viscous Rheology If the director is oriented parallel x 2 : General case; n notparallel x 2 :

6. Microstructures in Polycrystalline Materials during Deformation

7. Moving integration points We interpolate the nodal velocities using the shape functions to update the particle positions.  t is chosen “small” for accuracy purpose. The material history and stress rates are stored on particles.

8. Orthotropic folding ( click picture to play movie )

9. Example 1

10. Flow Alignment in Simple Shear

11. Extension with-and without yielding ( click picture to play movie ) …Nonlinear rheology, taken in the broadest sense, may be the single most important aspect of the behaviour of earth materials… Schubert, Karato, Olson, Turcotte From Outline of IMA Workshop Nonlin. Cont. Mech., Rheology and the Dynamo

12. Shear Histories simple shear and shear alignment with shear heating and temperature dependent viscosity

13. Shear-Heating: Director Field and Temperature Contours

14. Shear Alignment with Shear Heating and Temperature Dependent Viscosity

15. Director Models Liquid Crystals: de Gennes & Prost, 1972, 1993 Geophysics: U Christensen, 1984 (post – glacial rebound, mantle convection) Director Evolution (U CH.): Transforms as line element Present Model: Transforms as surface normal vector

16. Director Models Steady State The director evolution equation has a steady State solution in which the director is point-wise oriented normal to the velocity vectors. Solution maybe non-unique however……. Proof that is a particular solution for steady states:

17. Stability of Normal Director Solution Represented are 2 solutions:One assuming director normal to velocity and one where the 1 st 10 steps are run assuming normality and subsequent steps are integrated using full director evolution equation.

18. Convection with Shear Heating Full director evolution ; Di=0.25; Ra=1.2x10 6

19. Director Alignment

20. Degree of Alignment

21. Director Alignment in Convection Ra=0.5x10 6

22. Conclusion Rheology for layered materials as a basic unit (building stone) for more complex rheologies, modelling of crystallographic slip planes etc director orthogonal to velocity vector in steady state Orthogonal solution seems stable in convection Mean shear strain of approx 6 required for alignment in simple shear Examples include thermal coupling and influence thereof on alignment in simple shear, various convection studies Codes used: Fastflo, Ellipsis

23. Seismic Anisotropy

24. Convection with Ra =500.000 Isotropy Isoterms Anisotropy Velocity Field Stream function