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 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

Finite AnisotropyDirector evolution

n : the director of the anisotropyW, Wn : spin and director spinD, D’: stretching and its deviatoric part

)( kikjkjkiijn

ij DDWW

ij nin j

jn

iji nWn

Rotations

Spin of an infinitesimal volume element

1,2v 2,1v

2,11,22

1vv average

Spin of microstructure

n n

Undeformed ground state:

n.Wn.0v

v0n n

1,2

1,2

Anisotropic Viscous Rheology

If the director is oriented parallel x2:

122112

1111

2

,2

D

and

pD

S

,2 2222 pD

ijklijklSijij pDD '' )(22

General case; n notparallel x2:

Microstructures in Polycrystalline Materials during Deformation

Moving integration points

nodepnn

tp

ttp xNvtxx )(

We interpolate the nodal velocities usingthe shape functions to update theparticle positions.

t is chosen “small” for accuracy purpose.

The material history and stress ratesare stored on particles.

Orthotropic folding (click picture to play movie)

Example 1

Evolution of folding in anisotropic visco-elastic layer. Isotropic embedding material has viscosity 0.001, layer has shear viscosity 0.001, normal viscosity 1. Results are shown for perturbation to the director orientation with wavenumber 2q and 10q . Notice change in dominant mode

for 10q ;Deborah number De=VL

Flow Alignment in Simple Shear

Simple shear, same boundary conditions as in previous example however the director orientations are initially random: every particle of our advection scheme gets its own spatially randomly distributed director orientation; = angle between the x2 and the director orientation ranges between and -The top row are contour lines of . The deformation increases from (a) to (b). The green line (bottom) row, Figures (a)-(c) show initial and displaced particle positions. The parallel lines in (b) and (c) follow from the periodic boundary conditions

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, TurcotteFrom 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

maxv

maxT

10

12

22

maxT

ntdisplacemetop

alignment

maxv

maxT

12

22

5

0Tand0vv:0x

1TTand0v;1:1x:s.'c.B

1HlayershearofWidth

ntdisplacemetop

2,212

ref2122

maxv

0

190

p0

0p0

0

RT

Qexp108400

D

HVcPe79.0

HTc

VDi

0

Shear-Heating:Director Field and

Temperature Contours

10.1ntdisplacemeTop

Shear Alignment with Shear Heating and Temperature Dependent Viscosity

12

22

12

22

maxT

alignment

ntdisplacemetop

Director Models

Liquid Crystals: de Gennes & Prost, 1972, 1993Geophysics: U Christensen, 1984 (post –glacial rebound, mantle convection)

Director Evolution(U CH.):Transforms as line element

Present Model:Transforms as surface normal vector

nvnvn )()(t,

))()(t, nvnvn

Director ModelsSteady State

The director evolution equation has a steady State solution in which the director is point-wiseoriented normal to the velocity vectors.Solution maybe non-unique however…….

Proof that

0nnthat

followsit0vvandv

vcesin

if0)nv()nn(v

nvnv

i,kk,i

2,21,11

2

i,kki,kk,ik

ki,kk,ik

n

vn

nv 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 1st

10 steps are run assuming normality and subsequent steps are integrated using full

director evolution equation.

ntdisplacemetop

maxv

maxv

10

15

5

nDissipatio

maxT

61053.0Ra,27.0Di

dtvmax

Convection with Shear HeatingFull director evolution ; Di=0.25;

Ra=1.2x106

maxvx5.0

alignment

ndissipatiox5.0

dtvmax

Director Alignment

05.0dtvmax

05.0dtvmax

Degree of Alignment

nv1221 vnvn

sin

Director Alignment in ConvectionRa=0.5x106

)(

2 3

SNa

ThqR

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 =500.000

Isotropy

Isoterms

Anisotropy

Velocity Field

Streamfunction

)(

2 3

SNa

ThqR