Texture alignment in simple shear

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Nov 29, 2013 (3 years and 4 months ago)

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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.

0
div


f
σ
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

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5

Setting of the plastic viscosity

Experimental flow time of 5,2 s

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6

Numerical results on mortar

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7

Integration scheme

Unlike a standard FEM scheme
where the integration points are
worked out in advance for the
master element and weights
computed accordingly, the PIC
case, the particle positions are
imposed by the deformation only
the weights are unknown.


Balance between time
consumption, accuracy and
uniqueness of the solution
(negative weight)

Keep in mind that it is still
an approximation of an
iterative process

Constant terms





ep
n
1
p
p
2
Linear terms





ep
n
1
p
p
p
0
x
Quadratic temrs, etc…





ep
n
1
p
2
p
p
3
/
2
x
FEM

PIC














1
1
n
1
i
i
i
ep
f
d
f
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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.

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9

Solution de
référence

4 particules

Solution de
référence

Reference
solution

4 particles

16 particles

How many particles?



Initially 4
×
4 particles

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10

te
p
C


te
p
C


Termes

linéaires

Termes

constants

What condition on the weight?



Conditions to the linear terms

te
p
C


Constant terms

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11

Particle separation

Particle = Integration point


Concentrated representative v
olume

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12

Particle fusion

0
d
A
d


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13

Maxwell viscoelasticity

D
τ
τ






2
2
Deviatoric part

)
(
tr
p
K
p
e
D





Isotropic part

Integration scheme

t
t
t
t
e
t
t
t
t
t
t
)
(
e
e
ω
τ
τ
ω
τ
τ
τ










e
t
t
t
t
p
p
p
e






Deviatoric relaxation time





And volumic

e
K



Law

e
t
t
e
t
e
t
t
t
t
t
t
1

















σ
σ
σ
σω
ωσ
σ
σ





Jaumann derivative



16/12/2002

14

Viscoelastic oscillations

Shear

te
2
,
1
C
v

Constitutive relationship






























12
12
2
,
1
12
22
11
22
11
22
11
v
D
0
D
D
2
2
2
,
1
*
2
,
1
e
v
t
t
v
μ
η
relaxation

de

Temps
n
observatio
d'

Temps
D




Using

2
,
1
2
2
2
2
,
1
12
12
12
v
v
2
































Second order PDE



2
e
2
,
1
*
*
D
t
12
D
1
v
t
cos
B
t
sin
A
e
e
*







Solution

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15

10s
Δt
e

1s
Δt
e

0,1s
Δt
e

0,01s
Δt
e

10s
Δt
e

1s
Δt
e

0,1s
Δt
e

10s
Δt
e

1s
Δt
e

10s
Δt
e

10s
Δt
e

1s
Δt
e

0,1s
Δt
e

0,01s
Δt
e

0,006s
Δt
e

0,005s
Δt
e

10s
Δt
e

1s
Δt
e

0,1s
Δt
e

0,01s
Δt
e

0,006s
Δt
e

Stability/accuracy of the scheme



= 1,0s

h
0

= 1,0m


t = 0,01s

V

h(t)

Compression :

0

t

9s


V=0,1m/s

Relaxation :

A t=9s


V=0 m/s

Stability



Accurace


t
t
e



)
,
min(
01
,
0
t
e




16/12/2002

16

Cosserat theory

Stress approach

0
m
e
0
b
V
i
kl
ikl
j
,
ij
i
j
,
ij










Kinematic approach

c
k
ijk
j
,
i
ij
e
v




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17




























































































zy
zx
yx
xy
yy
xx
c
c
c
zy
zx
yx
xy
yy
xx
.
M
0
M
.
sym
0
0
0
0
0
0
0
0
0
0
0
0
2
c
d
2
M


Cosserat rheology in 2D

Bending viscosity

Where d is the internal

length of the material

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18

Flow of a Cosserat continuum

d/a=0

d/a=1/3

d/a=2/3

d/a=5/3

d/a=10/3

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19

Finite Anisotropy

Director evolution

n

: the director of the anisotropy

W, W
n

: spin and director spin

D
,
D
’: stretching and its deviatoric part


)
(
ki
kj
kj
ki
ij
n
ij
D
D
W
W






ij

n
i
n
j
j
n
ij
i
n
W
n


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20

Anisotropy (kinematic)

Evolution of the director

Evolution of the thickness of the layer

t
ω
n
n



Dn
n
T
h
h


n
3

c
3

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21



Couple stresses

Elastic:



2
1
2
2
3
1
X
u
h
m
s






Viscous:



2
1
2
2
3
1
X
u
h
m
s







2
1
X
u
c





X
1

t

h

F

16/12/2002

22

Virtual Power



















dV
X
u
h
E
E
s
standard
2
2
1
2
2
2
3
1


Requires continuously differentiable shape functions:

inconvenient !

Penalty formulation:









































dV
X
h
3
1
dV
u
P
2
dV
X
u
h
3
1
2
1
c
2
s
2
c
1
,
2
2
2
1
2
2
2
s






P
as
u
er
cov
re
to
hope
we
E
Since
c
1
,
2

16/12/2002

23







dV
:
P
P
V
c
ij
n
ij
c
ij
n
ij
c
ij
ij
ij
ij
int














P : penalty term

Anisotropy (C
0

reconstruction)





















V
V
i
S
i
i
i
i
V
i
i
i
V
ij
ij
ij
ij
dA
m
u
t
dV
m
u
b
dV
K
M
Principle of virtual power

ext
int
P
P



dV
P
2
P
V
mn
ij
ijmn
c
ij
ij
ij
ij
int











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24

Convergence of Penalty Method


Dimensionless velocity
)
/(
h
S
V
V



versus dimensionless
penalty parameter
S
P
P

/

; analytical solution (crossed);
numerical, full integration (broken line); numerical, one
point integration (solid line). Finite element model: ei
ght
by twelve four noded quadrilaterals; sixteen particles per
element. Periodic boundaries on the sides, i.e. velocities
and rotations are the same on both sides; if one particle
leaves the domain on one side it re
-
appears on the other
side.
2
/

S


,
t/h=0.2
; t=thickness of the individual
layers and h is the thickness of the shear layer. During the
calculation the director orientation is fixed at
n
=(1,0) , i.e.
the internal layering is always orthogonal to the x
2
=const.


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25

Anisotropic rheology

Constitutive relationship for the deviatoric stress

C
γ
τ









Δ
C
C
C
C
C
P
2
0
0
0
0
0
0
2
0
0
0
0
2
S
0
S
0
1
1
S
0
S
0
1
1
1
1
0
0
1
1
0
0
pen
ortho













































































2
2
2
1
S
0
n
n
4









2
1
2
2
2
1
S
1
n
n
n
n
2






with

16/12/2002

26

Flambement d’une couche anisotrope

Isotropic background medium with viscosity : 0,001 Pa.s

Anisotropic 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

5
D
e

Mühlhaus et al, 2002

16/12/2002

27

Simple Shear and Convection Problems

ij
'
kl
ijkl
S
'
ij
ij
p
D
)
(
2
D
2










Constitutive equations:





)
RT
Q
exp(
and
.
const
/
0
S






Temperature dependent viscosities:





Stress and Thermal equilibrium:





ij
ij
i
,
i
,
i
,
i
t
,
0
i
,
j
,
'
kl
ijkl
S
'
ij
D
)
kT
(
)
T
v
T
(
C
0
p
)
D
)
(
2
D
2
(













Non
-
dimensionalisation:





]
CT
/[
]
H
/
v
[
Di
]
v
/
H
/[
]
k
/
C
[
Pe
*
t
v
/
H
t
0
0
0
0
0
0
o






H
2
x


1
x





12
0
1
0
or
v
v
T
T
0
v
0
T
1
2
,


0
t
v
0

0
t
v
0



t
v
0
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28

Heat Equation cont.








ij
ij
i
,
i
,
i
,
i
t
,
D
Di
)
T
Pe
1
(
T
v
T
T








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



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


Reference viscosity

0
10
20
Pa


Density



Activation Energy Q= 180
-
550 KJ/mol


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

3
3
0
m
/
kg
10
5
.
4
5
.
2



16/12/2002

29

max
v
max
T
10


12
22
max
T



nt
displaceme
top
alignment
max
v

max
T


12
22
5
0
T
and
0
v
v
:
0
x
1
T
T
and
0
v
;
1
:
1
x
:
s
.'
c
.
B
1
H
layer
shear
of
Width
nt
displaceme
top
2
,
2
1
2
ref
2
12
2














max
v
0
19
0
p
0
0
p
0
0
RT
Q
exp
10
8400
D
HV
c
Pe
79
.
0
H
T
c
V
Di









10
Shear Histories simple shear and shear alignment with shear heating
and temperature dependent viscosity

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30

Shear
-
Heating : Director Field and Temperature Contours


10
.
1
nt
displaceme
Top


16/12/2002

31

Shear Alignment with Shear Heating and Temperature Dependent
Viscosity



12
22
max
T
alignment



nt
displaceme
top
n
v
n
v
1
2
2
1
v
n
v
n
)
,
sin(
alignment



Alignment=0 if n is parallel

to v and = 1 if n is orthogonal

(steady state!) to v.

Initial

configuration

Final (aligned)

configuration

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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.

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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