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

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 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 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, Today: Oriented materials and emergent anisotropy in simple shear and natural convection; thermal coupling in simple shear and …convection

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

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

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

Microstructures in Polycrystalline Materials during Deformation

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.

Orthotropic folding ( click picture to play movie )

Example 1

Flow Alignment in Simple Shear

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

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

Shear-Heating: Director Field and Temperature Contours

Shear Alignment with Shear Heating and Temperature Dependent Viscosity

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

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:

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.

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

Director Alignment

Degree of Alignment

Director Alignment in Convection Ra=0.5x10 6

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

Seismic Anisotropy

Convection with Ra = Isotropy Isoterms Anisotropy Velocity Field Stream function