Emergent Anisotropy and

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Emergent Anisotropy and Flow Alignment
in Viscous Rock by Hans Mühlhaus, Louis
Moresi, Miroslav Cada May 5-10
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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 DunckerHumblot , Berlin ,
  229-252
• Today Oriented materials and emergent anisotropy
  in simple shear and natural convection thermal
  coupling in simple shear and convection

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Finite Anisotropy
Director evolution
n the director of the anisotropy W, Wn spin
and director spin D, D stretching and its
deviatoric part
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Rotations
Spin of an infinitesimal volume element
Spin of microstructure
n
n
Undeformed ground state
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Anisotropic Viscous Rheology
If the director is oriented parallel x2
General case n notparallel x2
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Microstructures in Polycrystalline Materials
during Deformation
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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.
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Orthotropic folding (click picture to play movie)
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Example 1
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Flow Alignment in Simple Shear
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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
Extension with-and without yielding (click
picture to play movie)
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Shear Histories simple shear and shear alignment
with shear heating and temperature dependent
viscosity
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Shear-HeatingDirector Field andTemperature
Contours
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Shear Alignment with Shear Heating and
Temperature Dependent Viscosity
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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
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Director ModelsSteady 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
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Stability of Normal Director SolutionRepresented
are 2 solutionsOne assuming director normal to
velocity and one where the 1st 10 steps are run
assuming normality and subsequent steps are
integrated using full director evolution equation.
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Convection with Shear HeatingFull director
evolution Di0.25 Ra1.2x106
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Director Alignment
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Degree of Alignment
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Director Alignment in ConvectionRa0.5x106
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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

www.ned.dem.csiro.au/research/solidmech

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Seismic Anisotropy
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Convection with Ra 500.000
Isotropy
Anisotropy
Stream function
Isoterms
Velocity Field