16/12/2002 1

Texture alignment in simple shear

Hans Mühlhaus,Frederic Dufour and Louis Moresi

16/12/2002 2

Outline

I) Introduction

II) Numerical methods

III) Rheological models

IV) Applications

V) Conclusions and perspectives

16/12/2002 3

Problem

To model in large transformations a large range of materials at equilibrium.

0div fσ

16/12/2002 4

Idea of the Particle in Cell method (PIC)

• Eulerian finite element mesh

• Lagrangian particles used as integration points

• Time variables are stored on particles

• Updated Lagrangian formulation

16/12/2002 5

Setting of the plastic viscosity

Experimental flow time of 5,2 s

16/12/2002 6

Numerical results on mortar

16/12/2002 7

Integration scheme

Constant terms

epn

1pp 2

Linear terms

epn

1ppp 0x

Quadratic temrs, etc…

epn

1p

2pp 3/2x

FEM PIC

1

1

n

1iii

ep

fdf

16/12/2002 8

Forced convection

A reference solution is calculated over a 36800 node mesh, with Gauss

integration.

Parametric study over a 2300 node mesh with

particles regularly distributed and weighted

initially.

16/12/2002 9

Solution de référence

4 particules

Solution de référenceReference solution

4 particles

16 particles

How many particles?

Initially 4×4 particles

16/12/2002 10

tep C tep C

Termeslinéaires

Termesconstants

What condition on the weight?

Conditions to the linear terms

tep CConstant terms

16/12/2002 11

Particle separation

Particle = Integration point Concentrated representative volume

16/12/2002 12

Particle fusion

0dAd

16/12/2002 13

Maxwell viscoelasticity

Dττ

22Deviatoric part

)(trp

K

p

e

D

Isotropic part

Integration scheme

tttte

ttttt

t)(

e

e

ωττωττ

τ

e

ttt

t

ppp

e

Deviatoric relaxation time

And volumiceK

Lawett

et

ett

t

t

t

t1

σσσ

σωωσσσ

Jaumann derivative

16/12/2002 14

Viscoelastic oscillations

Shearte

2,1 Cv

Constitutive relationship

1212

2,112

221122112211

vD

0DD22

2,1*

2,1e

vtt

vμ

η

relaxation de Temps

nobservatiod' TempsD

Using

2,1

22

22,1121212 vv2

Second order PDE

2e

2,1**D

t

12 D1

vtcosBtsinAe e

*

Solution

16/12/2002 15

10sΔte 1sΔt e 0,1sΔte 0,01sΔt e

10sΔte 1sΔt e 0,1sΔt e

10sΔte 1sΔt e 10sΔte 10sΔte 1sΔt e 0,1sΔte 0,01sΔt e

0,006sΔt e 0,005sΔt e

10sΔte 1sΔt e 0,1sΔte 0,01sΔt e

0,006sΔt e

Stability/accuracy of the scheme

= 1,0sh0 = 1,0mt = 0,01s

V

h(t)

Compression :

0t9s V=0,1m/s

Relaxation :

A t=9s V=0 m/s Stability

Accurace

tt e

),min(01,0t e

16/12/2002 16

Cosserat theory

Stress approach

0me

0bVikliklj,ij

ij,ij

Kinematic approach

ckijkj,iij ev

16/12/2002 17

zy

zx

yx

xy

yy

xx

c

cc

zy

zx

yx

xy

yy

xx

.

M

0M.sym

00

00

0000

0000

2cd2M

Cosserat rheology in 2D

Bending viscosity Where d is the internallength of the material

16/12/2002 18

Flow of a Cosserat continuum

d/a=0

d/a=1/3d/a=2/3

d/a=5/3

d/a=10/3

16/12/2002 19

Finite Anisotropy

Director evolution

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

)( kikjkjkiijnij DDWW

ij nin j

jniji nWn

16/12/2002 20

Anisotropy (kinematic)

Evolution of the director

Evolution of the thickness of the layer

tωn n

DnnT

h

h

n3

c3

16/12/2002 21

Couple stresses

Elastic: 2

1

22

3

1

X

uhm s

Viscous: 2

1

22

3

1

X

uhm s

2

1

X

uc

X1

t hF

16/12/2002 22

Virtual Power

dVX

uhEE sstandard

2

21

22

2

3

1

Requires continuously differentiable shape functions:inconvenient !

Penalty formulation:

dVX

h3

1dVuP2dV

X

uh

3

12

1

c2

s

2c1,2

2

21

22

2s

Pasu

ercovretohopeweESincec

1,2

16/12/2002 23

dV:PPV

cij

nij

cij

nij

cijijijijint P : penalty term

Anisotropy (C0 reconstruction)

V V

iSiiii

Viii

V

ijijijij dAmutdVmubdVKM

Principle of virtual power extint PP

dVP2PV

mnijijmncijijijijint

16/12/2002 24

Convergence of Penalty Method

D im e n s io n le s s v e lo c i t y )/( hSVV v e r s u s d im e n s io n le s s

p e n a lt y p a r a m e t e r SPP / ; a n a ly t ic a l s o lu t io n ( c r o s s e d ) ;

n u m e r ic a l , f u l l in t e g r a t io n ( b r o k e n l in e ) ; n u m e r ic a l , o n e p o in t in t e g r a t io n ( s o l id l in e ) . F in i t e e le m e n t m o d e l : e i g h t b y t w e lv e f o u r n o d e d q u a d r i la t e r a ls ; s ix t e e n p a r t ic le s p e r e le m e n t . P e r io d ic b o u n d a r ie s o n t h e s id e s , i . e . v e lo c i t ie s a n d r o t a t io n s a r e t h e s a m e o n b o t h s id e s ; i f o n e p a r t ic le le a v e s t h e d o m a in o n o n e s id e i t r e - a p p e a r s o n t h e o t h e r s id e . 2/ S , t / h = 0 . 2 ; t = t h ic k n e s s o f t h e in d iv id u a l

la y e r s a n d h is t h e t h ic k n e s s o f t h e s h e a r la y e r . D u r in g t h e c a lc u la t io n t h e d i r e c t o r o r ie n t a t io n is fi x e d a t n = ( 1 , 0 ) , i . e . t h e in t e r n a l la y e r in g is a lw a y s o r t h o g o n a l t o t h e x 2 = c o n s t .

16/12/2002 25

Anisotropic rheology

Constitutive relationship for the deviatoric stress Cγτ

ΔC

CCCC

P2

00

00

0020

0002

S0S011

S0S011

1100

1100

penortho

22

21S0 nn4

21

2221S1 nnnn2

with

16/12/2002 26

Flambement d’une couche anisotrope

Isotropic background medium with viscosity : 0,001 Pa.sAnisotropic layer of normal viscosity 1 Pa.s

And tangenial viscosity 0,001 Pa.s

Initial perturbation

of the director’s

orientation

Change of major mode for 5De

Mühlhaus et al, 2002

16/12/2002 27

Simple Shear and Convection Problems

ij'klijklS

'ijij pD)(2D2

Constitutive equations: 

)RT

Qexp(and.const/ 0S

Temperature dependent viscosities: 

Stress and Thermal equilibrium: 

ijiji,i,i,it,0

i,j,'klijklS

'ij

D)kT()TvT(C

0p)D)(2D2(

Non-dimensionalisation: 

]CT/[]H/v[Di

]v/H/[]k/C[Pe*tv/Ht

0000

00o

H2x

1x

12010 orvvTT

0v0T 12, 0tv0

0tv0

tv0

16/12/2002 28

Heat Equation cont.

 

ijiji,i,i,it, DDi)TPe

1(TvTT

 

C is the heat capacity; 800-1000 W/(kg K)

k is the thermal conductivity; 2.3-3.5 W/(m K)

Reference viscosity Pa

Density

Activation Energy Q= 180-550 KJ/mol

Universal Gas Constant R=0.00831J/(mol K)

330 m/kg105.45.2

16/12/2002 29

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

Qexp10

8400D

HVcPe

79.0HTc

VDi

10

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

16/12/2002 30

Shear-Heating : Director Field and Temperature Contours

10.1ntdisplacemeTop

16/12/2002 31

Shear Alignment with Shear Heating and Temperature Dependent Viscosity

12

22

maxT

alignment

ntdisplacemetop

nvnv 1221 vnvn

),sin(alignment

Alignment=0 if n is parallel to v and = 1 if n is orthogonal(steady state!) to v.

Initialconfiguration

Final (aligned)configuration

16/12/2002 32

Conclusions

Benchmark for the Particle in Cell method

16 particles initially for a good integration

Constraints on the weight to the linear terms

Appetite of 0,8

Developing/implementing new rheologies

Cosserat continuum

Viscoelasticity

Anisotropic model (classical or in a Cosserat context)

Bingham’s law for mortar

Benchmarks were successfull in the context of comparison with theory of with other numerical methods.

16/12/2002 33

Conclusions

Applications

Performant tool to study geological folding

Promising first steps on the study of fresh concrete flow

Drawbacks of the method

Traction boundary conditions

Diffusion of the interface by separation of the particles

Uncertainty on the quality of the numerical integration

Expertise needed