Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 1/22

Organization (S): EDF-R & D/AMA
Handbook of reference
R5.04 booklet: Nonlocal modelings
Document: R5.04.11
Model of Rousselier with gradient of variables
interns

Summary:

One presents here the model of Rousselier in great deformations in a nonlocal version i.e.
introducing gradients of variables intern in order to take into account strong space variations of
mechanical fields. One activates the nonlocal formulation of the model of Rousselier by one of modelings
“X_GRAD_VARI” of command AFFE_MODELE of key word MODELE. As for the model even, it is available
in command STAT_NON_LINE via the key word RELATION = “ROUSSELIER” under
key word factor COMP_INCR and with the key word DEFORMATION = 'SIMO_MIEHE.
This model is established for three-dimensional modelings (3d_GRAD_VARI), axisymmetric
(AXIS_GRAD_VARI) and in plane deformations (D_PLAN_GRAD_VARI).

One presents the writing and the digital processing of this model.
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 2/22

Count

matters

1 Recall on the theory of the models with gradient ....................................................................................... 3
1.1 Construction of the models with gradient .............................................................................................. 3
1.2 Discretization in time ................................................................................................................... 4
1.3 Space discretization by finite elements ........................................................................................ 4
1.4 Calculation of the variables intern at the points of gauss ......................................................................... 5
2 Application to the model of Rousselier .................................................................................................... 6
2.1 Some notations of the model of Rousselier ................................................................................. 6
2.2 Continuous model ................................................................................................................................ 7
2.3 Model discretized ............................................................................................................................. 8
2.4 Processing of the singular items .................................................................................................... 10
3 numerical Resolution .......................................................................................................................... 11
3.1 Expression of the model discretized ................................................................................................... 11
3.2 Resolution of the nonlinear system ............................................................................................... 12
3.3 Course of calculation ................................................................................................................... 13
3.4 Resolution of the functions to cancel ............................................................................................... 14
3.4.1 Hight delimiters and lower if S (0) > 0 ..................................................... 14
3.4.2 Hight delimiters and lower if S (0) = 0 ..................................................... 15
3.4.3 Hight delimiters and lower if S (0) < 0 and xs not solution ......................... 16
3.5 Form of the tangent matrix ................................................................................................ 17
3.5.1 Elastic case ........................................................................................................................ 18
3.5.2 Singular case ......................................................................................................................... 18
3.5.3 Regular case .......................................................................................................................... 19
4 Relation `ROUSSELIER `........................................................................................................................ 21
5 Bibliography ........................................................................................................................................ 21

Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 3/22

1
Recall on the theory of the models with gradient

The models with gradient presented here were developed by E. Lorentz [bib1] in order to be able to describe it
material behavior requested by strong gradients of the mechanical fields which appear
in the damaged zones or in the vicinity of geometrical singularities. Indeed, in
case of strong gradients, the behavior of a material point is not independent any more of its entourage
but depends on the behavior of its vicinity, from where the introduction of gradients into the models.
From a numerical point of view, the calculation of a structure with a local law of damage
traditional watch which the damaged zone always locates on only one layer of finite elements
and thus that the response of the structure depends on the adopted grid: the models with gradient mitigate it
problem.
In what follows, we make a short recall of this theory. One will find in [R5.04.01] details
thorough on this theory.

1.1
Construction of the models with gradient

This formulation is restricted with generalized standard materials. The state is described there by
deformation, of the internal variables has and their associated gradient a:

has = has éq 1.1-1

According to the formalism of generalized standard materials, the data of the free energy
(, has, has)
and of the potential of dissipation (&a, has &)
(for the choice of these two energies, one will be able
to refer to [bib1]) allow to deduce from them the laws from state and the laws of evolution:







=
, A = -
, A = -









éq 1.1-2

has


has

(,
WITH A)
(&a, has &)















éq 1.1-3
If one calls F the threshold of elasticity associated with the potential (&a, has &)
, the preceding equation is
equivalent to:
F
F
has & = &
, has & = &










éq 1.1-4
With

With
The problem here is that the variables are not independent any more and are bound by the constraint not
local [éq 1.1-1] so that one is not sure to check:
F
&a = &
= has & éq 1.1-5
With
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 4/22

One then proposes to forget the assumption of normal flow in each point of the structure all in
preserving the formalism of generalized standard materials but on the scale of the structure, where
variables of state are now the field of deformation and the field of internal variables A. One
thus defines the total free energy and the total potential of dissipation:




F (, has) = ((X), has (X), has (X))D X éq 1.1-6

D (a&) = (a& (X), a& (X))D X éq 1.1-7


The relation of behavior generalized is written now:



=

F, A = -
F, AD (a&) éq 1.1-8

has


1.2
Discretization in time

While being based on the assumption of convexity compared to A of the potentials
F and D and by adopting one
diagram of implicit Euler, the temporal discretization of the preceding problem [éq 1.1-8] is reduced to
resolution of a problem of minimization relating to the increment has fields of variables
interns. This problem is written for behaviors independent of time:
Min [F (, A

+ has
) + D (has
)] éq 1.2-1
has

where -
A is the field of internal variables at the previous moment.

1.3
Space discretization by finite elements

To solve the problem of minimization [éq 1.2-1], one carries out a space discretization by
finite elements of the fields of internal variables by means of the nodal unknown factors which one will note A.
has (X) = NR (X
K
) K
With, has (X) = NR (X
K
) K
With
éq
1.3-1
node
node
where Nk and Nk are related to form and their gradients associated with the node K, respectively.
To simplify the writing, one will pose:
R
R
R
R
has (X) = B (X) A

with

has = (has, has and

)
B = (NR, NR) éq
1.3-2
The equation [éq 1.2-1] is written then:
R
R
Min G ((B A)
G
+ (BgA)) éq 1.3-3
With gauss
in which G indicates the weight of integration of the point of Gauss G.
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 5/22

The introduction of new unknown factors R which represent the values and the gradients of the variables
interns at the points of gauss allows to transfer nonthe linearities at the local level and to divide
resolution of the equation [éq 1.3-3] in a linear total part (on the structure) and a local part
nonlinear (at the points of integration). The problem of minimization is written then:

Min

R

R
R
R
G (

(G) +
(G - -
G) éq 1.3-4
With
R
,
gauss

R


G (B A
G
-

0
=
)
G

By dualisation of the constraint, one builds increased Lagrangien of the problem [éq 1.3-4] for
to return to a problem without constraints:
Max Min L
R (A, R, µr) éq 1.3-5
µr
With, R
with
R
R R

R
R
R -
R R
R 2
R R

L (A, µ) =
(
) +
(
-) + B A - R
G
G
G
G
G
G R + µ (B A -
R)
NR

2
G
G
G
G
G

R
éq 1.3-6
R r2
R R
R
> 0
and

X R = X NR
.
xr
.
G
Ng
R
R
The positive matrix NR definite is introduced into the standard so that the coefficient of penalization
R is adimensional. This matrix is selected like a diagonal approximation of the derivative
second (cf [R5.04.01] for more detail):
R
R
2
=
NR Diag

R R éq 1.3-7



This problem is then solved by a method of Newton to solve the primal problem (calculation
of A and R) and a method BFGS with linear search Wolfe to solve the dual problem
(calculation of the multipliers of Lagrange).

1.4
Calculation of the variables intern at the points of gauss

At the time of the resolution of the local problem, one seeks to minimize the equation [éq 1.3-6] compared to R,
with A and µr fixed what is equivalent to:
R
R R


R
R R
L
Min R (fixed
With
, fixed
µ
)
fixed
G
-
+ µ
+ RN (
fixed
B WITH
- R)
(R
)
G
G
R
R
G
G
G
G
G éq
1.4-1


G
G

With convergence, the third term in the member of left of the expression above becomes null
(the constraint is carried out) and the multipliers of Lagrange µr G seem a force then
R
complementary thermodynamics resulting from the nonlocal condition B
R
G A - G = 0.
From a practical point of view, to write in an incremental way the model of behavior not
room, one will write classically the equivalent of the equation [éq 1.4-1]:
R
F
R
= &

with

Ar
R
= (R
With, R
With) éq
1.4-2
R

With

Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 6/22

with
R R R
R
R
R R
R
R


With = A + µ + RN (BA -
With

with
)
= (,
WITH A) = -
éq
1.4-3
R

where F is the threshold of elasticity associated with the potential of dissipation, calculated by transform of
Legendre-Fenchel, and R
With and R
With the thermodynamic forces associated the internal variables and
, respectively. It is noted here that these two thermodynamic forces must be corrected,
on the one hand, by the multipliers of Lagrange associated with the nonlocal constraint =, and
in addition, by a measurement (balanced) of the variation enters the fields at the points of gauss R = (,)

and nodal field A.

2
Application to the model of Rousselier

We now describe the application of this theory to gradient to the model of Rousselier
(cf [R5.03.06] for more detail on this model).

2.1
Some notations of the model of Rousselier

One points out below some definitions and notations used in the model of Rousselier.

F: tensor gradient which makes pass from the initial configuration 0 to the current configuration (T)
F p: “plastic” tensor gradient which makes pass from configuration 0 to the slackened configuration
R
Fe: “elastic” tensor gradient which makes pass from the configuration R to (T)
E p
F = F F éq 2.1-1
J: variation of volume
J = det F éq 2.1-2
Be: left tensor eulérien of Cauchy-Green of elastic strain
Be
FeFeT
=












éq 2.1-3
G p: Lagrangian tensor of plastic deformation
G p
F pTF p
=
-
(
) 1 éq 2.1-4
Be
FG pFT
=












éq 2.1-5
E: tensor of the deformations used in the model of Rousselier
1
E = (
E
Id - b) éq 2.1-6
2
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 7/22

D: rate of deformation
p
D: rate of plastic deformation
p
1
p T
D - FG & F éq 2.1-7
2
: constraint of Cauchy
: constraint of Kirchhoff
=
J















éq 2.1-8
S: tensor of the constraints used for the model of Rousselier
E
= S B éq 2.1-9
R = - isotropic a: work hardening
p: cumulated plastic deformation
F: porosity
1
F & = (1 - F)
p T
tr-
G
F & F éq 2.1-10
2


F: criterion of plasticity of the model of Rousselier
tr S
F (S, A) = eq
S + Df exp
1
+ A (p) -




y
éq
2.1-11

3 1
where is limited there from elasticity and 1, D two parameters materials specific to this law.

2.2 Model
continuous

To preserve a simple model, one will be satisfied to introduce, to control the modes of
localization poro-plastic, a quadratic term in gradient of cumulated plastic deformation p
in the free energy of the model of local Rousselier. As for the potential of dissipation, there remains unchanged
compared to the local version.
p
2
(
1
2
2L D R
E, p, p
~ ~
B
) =
[K (tre) + µ2e:E] + R (U) du+
p p
. éq
2.2-1
2
0
13 D p


D

p
tr p
p
p
2
(D, p&, &p)
= p & + tr D ln
- 1 + I + (tr D) + I + (p & -
p
D)
y
1


IR
IR
éq
2.2-2
Df p

&
3 eq

This potential of dissipation corresponds to the criterion of plasticity [éq 2.1-11].
B
L is the length characteristic of the material which corresponds to the average distance between two
inclusions, privileged sites of germination and nucleation of cavities.
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 8/22

2.3 Model
discretized

To adopt a purely implicit algorithm to integrate this law of led behavior, on the one hand,
the resolution of a rather complex nonlinear system any more, and in addition, does not allow to express it
problem like the minimization of a functional calculus. This is why one prefers to deal with manner
clarify the variation according to porosity in the potential of dissipation [éq 2.2-2] as well as the variation
according to p of the quadratic term out of p of the free energy [éq 2.2-1]. For the other terms, one
employ a diagram of implicit Euler. It will be noted that the discretization of the rate of plastic deformation
p
D is expressed directly according to the elastic strain E:




p
1
p T
D - FG & F = - 1 [E
p T
B - FG F] = 1
1
E - [
p T
Id - FG F]
Tr
= (E - E)/T
éq 2.3-1
2
2 T

T
2

1 4
4 2 4
4 3

T

R
E


So that the free energy and the discretized potential of dissipation are given by the expressions
following:
p
2
(
1
2
2L D R
E, p, p) = [K (tre) + µ e~
2
e~
: ] + R (U)
B
-

+
(p p
) p
. éq 2.3-2
2
0
13 D p


tr E


2
(,
E p
) = p
+ tr E
ln
- 1 + I + (tr E
) + I + (p
-
E
)
y
1

-

IR
IR
éq
2.3-3
Df
p

3
eq

with
-
T = T - T,
tr
E
= E - E,
-
p = p - p and
-
p = p - p éq
2.3-4
-
Q is the quantity known at the previous moment -
T.
In accordance with the paragraph [§1.4] [éq 1.4-1], the integration of the model of nonlocal Rousselier
express yourself like the minimization of the following functional calculus:

Tr
R
min
2
R
(
R
R
R R
R R
E +
-
E p
,
+ p) + (E
, p) + P - p rr +µ (P - Pr)

éq.
2.3-5
E, p
NR


2


What is equivalent to [éq 1.4-1] and [éq 1.4-2]:





F

E =
S = -

E

S


F

(,
E p) p =
éq 2.3-6
R
R
R
R
R R


R
With
R
With = A + µ + RN (P - Pr)


R
To = 0

Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 9/22

R
R
According to the equation [éq 1.3-7], the matrix of weighting NR is worth:
1 0
0
0
2

2
R
R

0 L
0
0
-
C
-
D R
-
2
4 2
=
NR Diag
H

=
R R
2
with

H

=
(p)
and
L =
L


éq
2.3-7
p
p
0 0 L
0
D
C
p
13 B


C

2
0
0
0

C
L
In all these equations, one adopted the following notations:
ruffle = (, AA) = (- R, R)

- éq 2.3-8
R R
With = (R
With, R
With)
éq 2.3-9
Pr = (p, p)
éq 2.3-10
R R
P = LP = (P, P)
éq 2.3-11

The vector Pr represents the cumulated plastic deformation and its gradient, calculated at the points of
Gauss while P represents the plastic deformation calculated with the nodes.

µr = (µ, µ)
éq 2.3-12

The whole of the equations to be solved is thus the following:

Equations of state:
S [
-
= K tr ED + µe~
2] éq 2.3-13
Ar -
= R (p) µ
+ + Rh (P - p) -
= R (p) - C p +
1
c2
éq
2.3-14
R
- 2
- 2
With -
= H L p
C +µ + Rh LLC (
P - p) -
= C p + C = 0
3
4
éq
2.3-15
Laws of flow:
tr S
tr e=
-
Pd F

exp











éq 2.3-16
1

3
~ 3
s~
e=
p














éq 2.3-17
2
seq
c4
p=















éq 2.3-18
3
C
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 10/22

Condition of coherence:
(
S
F S, Ar)
-
tr
= S + D F exp
+ Ar
eq
- y F 0 p 0 FP 0
1
=



éq 2.3-19
3
1

Definition of the various coefficients:
C = -
Rh
0
- 2
2
1
, C = µ C P
2
+ 1, C = 1
(+ R) H
0
3
C
L and c4 = µ + 1
C C
L
P éq 2.3-20

2.4
Processing of the singular points

The expression of the normal cone to which the direction of flow belongs is licit only at the points where
the criteria are derivable, i.e. if seq 0. While proceeding classically by a prediction
rubber band followed by a plastic correction only if necessary, one can be satisfied to examine
singular points confined on the border of convex of elasticity, i.e. points such as:
tr S
~
1
-
R
y
S = 0
and

D F

exp
+ A =







éq 2.4-1
1

3
The normal cone with convex of elasticity in such a point is the whole of the directions of flow which
carry out the problem of maximization according to:
sup [S: p
D + Ar p & + R
p
With &p - (D, p&, &p)]
p
D, p&, &p
1

p
R
y
p
tr p
D

=
sup
tr str D + A p

éq 2.4-2
& - p & - tr D ln
1
-
1
p
D, p&, p
3
&
D F p&



-




tr p
D 0
p&-
2
L &p
D p 0
B
- 3
eq
They are the directions of flow (Dp, &p p
, &) characterized by:

tr S
p

tr D = D F p&


exp




3 1 éq 2.4-3
2 p
Deq p

&
3
Thus, in a singular point of the border of the field of elasticity, the increments of the variables
interns (E
, p
, p) check simply:

tr S
-

tr E = D F p


exp




3 1 éq 2.4-4
2
(E) eq p
3
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 11/22

3 Resolution
numerical

3.1
Expression of the discretized model

R
Knowing µr and P, elastic strain -
E, cumulated plastic deformation -
p, it
gradient of the cumulated plastic deformation -
p and displacements U and U, one seeks with
to determine (,
E p, p, F)

.
Displacements being known, gradients of the transformation of 0 with -, noted F, and of -
with, noted F, are known.

The system of equations to be solved is as follows:
-
F = FF éq 3.1-1

J = det F éq 3.1-2


J = éq 3.1-3

E
= S B éq 3.1-4

Be = Id - E
2 éq 3.1-5

Tr
1
E = [Id - F
{Id - E
2} T
F

] éq 3.1-6
2
Equations of state:
S [
-
= K tr eId + µe~
2] éq 3.1-7

Ar = - R (p) - C p +
1
c2 éq 3.1-8

Ar=- p
3
C + c4= 0 éq 3.1-9
Definition:
4
C = -
Rh
0
- 2
2
2
1
, C = µ C P
2
+ 1, C = 1
(+ R) H
0
3
C
L, c4 = µ + 1
C C
L
P,
2
C
L =
L éq 3.1-10
13 B
Thereafter, one expresses the laws of flow and the criteria of plasticity directly according to
tensor of the elastic strain E.
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 12/22

Laws of flow:
Tr
-
K tr E
tr E - tr E
p

= Df exp (-
) éq
3.1-11
1


~
Tr - 3
~
E
E
p

D

solution

if
gulière
~
2
E
E =
eq
éq
3.1-12

2
0


and

p (E)
singuli

solution

if
era

eq
3

c4
p=












éq 3.1-13
3
C
Condition of coherence:


-
K tr E
2µe + D F exp -
+ Ar
eq
- y
(
1




D

solution

if
gulière
R



F S, A)
1
=



-
K tr E
éq
3.1-14
D F exp
R
y
1

-
+ A -





singuli

solution

if
era


1

with F 0 p 0 FP = 0
Porosity:
The law of evolution of porosity is treated same manner as in the model of Rousselier in
local version. One obtains (cf [R5.03.06] for more detail):
F = 1 (1 - F exp (treTr
0)
- tr E) éq
3.1-15
where f0 is initial porosity.

3.2
Resolution of the nonlinear system

The integration of the law of behavior is thus summarized to solve only the equations
[éq 3.1-11], [éq 3.1-12] and [éq 3.1-14] (the equation [éq 3.1-13] gives p directly since c4 and 3
C
are known). Once determined p and E by the whole of these three equations, one deduces some
constraint S by the equation [éq 3.1-7], the constraint of Cauchy by the equations [éq 3.1-3] and
[éq 3.1-4] and porosity F by the equation [éq 3.1-15].
It is noticed that the three equations to be solved are identical to those of the model of Rousselier
room where one changed only thermodynamic force A by R
A. The resolution is thus
identical to this model. For this reason, we give only the broad outline. For more
detail, the reader will refer to the document [R5.03.06].
If one poses:
= tr E - tr Tr
X
E 0 éq 3.2-1
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 13/22

then, the equation [éq 3.1-11] is written:
K tr Tr
E
p
(X) 1
K X
= X exp
with G =
-
D F

exp -
éq 3.2-2
G


1

1

After some calculations, the equation [éq 3.1-12] becomes:




3 p (X)
1
~Tr
E
réguliè

solution

if
Re
~

Tr
2

E (X) =
eq
E


éq
3.2-3

2
0


and

p
Tr
E
singuli

solution

if
era

eq
3

Lastly, if one poses:
Kx
S (X) = R (p (X))


+ y + 1
C p (X) - c2 - 1G


exp -

éq 3.2-4

1
where S (X) is a continuous and strictly increasing function X, then the condition of coherence
[éq 3.1-14] is written (while using [éq 3.2-3]):
Tr
(
E
p
X
F X) µ
2 eq - 3
µ - S ()
0
R

solution

if
égulière
=
éq 3.2-5
S (X) =
0

singuli

solution

if
era

One thus brings back oneself to solve this scalar equation in X. Variable X is positive or null X 0
to guarantee a positive cumulated plastic deformation and the elastic solution is obtained for x=0.

3.3
Course of calculation

The general step to determine X is as follows:

1) One seeks if the solution is elastic
· calculation of F ()
0
· if F ()
0 < 0, then the solution of the problem are the elastic solution Sol
X
= 0
· if not one passes into 2)
2) If
S ()
0 > 0, the solution are plastic and regular
· one passes into 4)
3) If
S ()
0 < 0, one seek if the solution is singular
· one solves S (S
X) = 0
2
· if S
X checks the inequality
S
Tr
p
~

Ground
eq
E, then the solution is singular
S
X
= X
3
· if not, S
X is a lower limit to solve F (Sol
X
) = 0, one pass into 4)
4) The solution is plastic and regular
· one solves F (Sol
X
) = 0
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 14/22

3.4
Resolution of the functions to be cancelled

To solve the equations S (X) = 0, F (X) = 0, one employs a method of Newton with terminals
controlled coupled to dichothomy when Newton gives a solution apart from the interval
of the two terminals. One now presents the determination of the terminals for each case
precedents.

3.4.1 Hight delimiters and lower if S (0) > 0

One solves:

Tr
µ
Kx
2 eq
µe - S (X = 3
)
X exp (
)
F (X) = 0

4
42
1
4
43 G




1

F
1 4
4 2 4
4 3 éq
3.4.1-1
F ()
0 > 0

1
p

F ()
0
1
> 0
where the function p
(X) is continuous, strictly increasing and null at the origin and the function F ()
1 X is
continue, strictly decreasing and strictly positive at the origin (see [Figure 3.4.1-a]).

One poses:
Tr
Kx
F = 2µeeq - R (X) - y - C p + C + G exp (-
) then F
(X) < F (X)

X 0
1
1
2
1
2
1
éq 3.4.1-2
1
4
4
4
4
4
2
4
4
4
4
4
3
1
f2
where the function F ()
2 X are continuous, strictly decreasing. Two cases will be considered.
Case where F (0)
0
2
>
In this case, the successive resolution of the equations:
Inf
Inf
F (p
2
) = 3µ p










éq 3.4.1-3
to deduce p from it
, then
Inf
Inf
Kx
Inf
X
exp (
) = G p









éq 3.4.1-4
1
to deduce X from them a lower limit Inf gives
X
.

Note:

In Code_Aster, the routine rcfonc solves the equation corresponding to the solution of the model to
isotropic work hardening and criterion of Von Mises, i.e. el
- R (p) - = 3 p
eq
y

µ. One provides
in input of this routine el
eq, the Young modulus E and the Poisson's ratio. If one poses
el
=
1
(
2 +)
2 eTr
-
Inf
eq
µ eq - 1
C p + c2 and E =
[3µ+ 1c], the function
Inf
F (p
2
) = 3µ p


3
bring back to solve an equation of the el type
- R (p) - = 3 p
eq
y

µ.
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 15/22

Case where F (0)
0
2

In this case, the lower limit is taken equalizes to zero: Inf
X
= 0.

The upper limit Sup
X
is such as:
Sup
Inf
KxSup
Sup
G
3
p

µ
= F (X) X
exp (
)
F (xInf)
1
=

éq
3.4.1-5
3µ 1
1
Kx
The equation of type X exp (
) = constant is solved by a method of Newton.
1


3
Sup
Inf
f1 (X)
p

µ (X)
3
p

µ (X) = f1 (X)
f2 (X)
3
p

µ (Inf
X
) = f2 (Inf
X
)
X
Inf
X
Ground
X
Sup
X

Appear 3.4.1-a: graphic Représentation of the hight delimiters and lower

3.4.2 Hight delimiters and lower if S (0) = 0

The system to be solved is as follows:

X
Kx
-
Kx
S (X) = 0
R p + exp (
) + y + C p
1 - c2 = G
1 exp (-




)


G
1
1 éq
3.4.2-1
S (0) < 0


-
-
R (p) + y + C p
1
- c2 < G
1
The part of left is a continuous function, strictly increasing X, the part of straight line is one
continuous, strictly decreasing of X and strictly positive function in the beginning.
Two cases must be considered.
Case where R (-
p) + y +
-
C p - C > 0
1
2
:
Using the properties of the two functions, a chart (cf [Figure 3.4.2-a]) of these
functions shows that the upper limit Sup
X
is such as:
KxSup
Sup
G
1
G exp (-
) = R (-
p)


-
1

+ y + 1
C p - c2
X
=
1
log


K

-
-
1
R (p)


+ y + 1
C p - c2
éq 3.4.2-2
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 16/22

The lower limit Inf
X
is such as:
KxInf
Sup
Sup
1
G exp (-
) = R (p) + y + 1
C p
- c2
1
+ éq
3.4.2-3


Inf
1

X
=
1
G
log
K

Sup
Sup
R (p
)


+ y + 1
C p
- c2
Case where R (-
p) + y +
-
C p - C 0
1
2
:
In this case, one will take for the lower limit:
Inf
X
= 0 éq 3.4.2-4
The successive resolution of the two equations:
Sup
-
Sup
Sup

Kx
G
Sup
1
+ c2 - C p
1
- R (p
) - y = C p

1
and
Sup
X
exp (
) = G p

éq 3.4.2-5
1
allows to deduce p from it
then X to give an upper limit Sup
X
. To solve the equation of
left above, it is enough (see remark of the preceding paragraph) to use the routine rcfonc in
1
(
2 +)
posing el

-
eq = 1
G - 1
C p + c2, E =
1
C and = 1.
3


R
G
(X) + y +1 p - 2c
1



Kx
1 G


exp -



R (p) +
-


1
y + 1
C p - c2
Inf Ground
Sup
X
X
X
X

Appear 3.4.2-a: graphic Représentation of the hight delimiters and lower

3.4.3 Hight delimiters and lower if S (0) < 0 and xs not solution

The following system is solved:

Tr
µ
3
Kx
2µeeq - S (X) =
X exp (
)
4
42
1
4
43 G
1

1 4
4 2 4
4 3
F (X) = 0

f1


p

S (0) < 0



F (0)
1
> 0

éq
3.4.3-1

S

S (X) = 0
S

Tr
µ
3
S
Kx
2µeeq =
X exp (
)
G

1

Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 17/22

The Sol solution
X
is such as S (Sol
X
) > 0.
For the lower limit, one takes Inf
S
X
= X. Being given properties of the functions 1f
(strictly decreasing) and 3µ p (X) (strictly increasing), the upper limit Sup
X
is such
that (cf [Figure 3.4.3-a]):
Sup
Sup
Kx
G
2
Tr
X
exp (
) =
eq
E éq
3.4.3-2
3
1
This equation is solved by a method of Newton.

S (X) < 0 S (X) > 0
2µeTr
eq - S (0)
S (X) = 0

Kx
Tr
X exp (
)
2µeeq
G
1

X
Inf
S
Ground Sup
X
= X X X

Appear 3.4.3-a: chart of the hight delimiters and lower

3.5
Form of the tangent matrix

The resolution of the primal problem (calculation of P and Pr) by a method of Newton, requires calculation
following tangent matrix:
p
p
R


Pr = P
P
H
éq 3.5-1
P
p
p
P
P



Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 18/22

3.5.1 Case
rubber band

It is pointed out that the elastic solution is given by:
-
p = p éq 3.5.1-1

c4
p =













éq 3.5.1-2
3
C
what gives for the tangent matrix:
p

p

= 0
= 0
P

P
I

éq
3.5.1-3
p
I
p
I
R
= 0
=
ij
P

P
J

1+ R

3.5.2 Case
singular

The singular solution is given by the equations:
-
X Kx/1
p = p +
E










éq 3.5.2-1
G

c4
p =













éq 3.5.2-2
3
C
with X which checks:
S (X) = R (p)
- Kx/1
+ C p - C + y - Ge
= 0
1
2
1
éq
3.5.2-3
and
C = µ C P
- 2
2
+ 1, c4 = µ + Rh C
L
P éq
3.5.2-4
The linearization of this system gives:
1
Kx
p
=
1
(+
) eKx/1 X
éq 3.5.2-5
G
1
4
1
4
2 3
coeff 1


R
p =

P éq 3.5.2-6
1+ R

S (X) = (H + c)
- Kx/1
p - C P + KGe
X = 0
1
1
éq
3.5.2-7
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 19/22

By replacing p
in the equation above, one obtains:
C e-Kx/1
X
=
1
P
éq
3.5.2-8
coeff 1 (H + c1) + KGe-2Kx/1
what gives for the expression of p
:
1
p
=
C coeff 1 P
1
éq 3.5.2-9
(H + C coeff 1
1
+ KGe-2Kx/1
)
1
4
4
4
4
2
4
4
4
4
4
3
coeff 2
maybe for the tangent matrix:
p

C coeff 1
p
1

=

= 0
P

coeff 2
P
I
éq
3.5.2-10
p
I
p
I
R
= 0
=
ij
P

P
J 1+ R

3.5.3 Case
regular

The regular solution is given by the equations:
-
X Kx/1
p = p +
E








éq 3.5.3-1
G

c4
p =











éq 3.5.3-2
3
C
X checks:
F = 2µeTr
eq - 3
µ p (X) - (
S X) = 0 éq
3.5.3-3
The linearization of the whole of these equations gives:
p
= coeff E
1 Kx/1 X
éq 3.5.3-4


R
p =

P éq 3.5.3-5
1+ R

F = - 3µp (X) - (
S X) = 0 éq
3.5.3-6

S (X) = (H + C p
1) - C P

1
+ KGe-Kx/1 X
éq 3.5.3-7
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 20/22

While replacing in the equation [éq 3.5.3-6], the expression of p
[éq 3.5.3-4], one obtains for X
:
(coef1 (h+c1+ µ3 +KGe-2Kx/1
)
) eKx/1 x=c P
1


éq. 3.5.3-8
1
4
4
4
4
4
4
2
4
4
4
4
4
4
3
coeff 2
One finds then:
C coeff 1
p
= 1
P
éq 3.5.3-9
coeff 2


R
p =

P
éq
3.5.3-10
1+ R
what makes for the tangent matrix:
p

C coeff 1

p
1

=

= 0
P

coeff 2
P
I
éq
3.5.3-11
p
I
p
I
R
= 0
=
ij
P

P
J 1+ R

Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 21/22

4 Relation
`ROUSSELIER `

One activates the nonlocal formulation of the model of Rousselier by one of modelings
“X_GRAD_VARI” of command AFFE_MODELE of key word MODELE. As for the model even, it is
available in command STAT_NON_LINE via key word RELATION:
“ROUSSELIER” under the key word factor COMP_INCR and with key word DEFORMATION:
“SIMO_MIEHE”.
The whole of the parameters of the model is provided under the key words factors `ROUSSELIER `or
`ROUSSELIER_FO `and `TRACTION `(to define the traction diagram) command DEFI_MATERIAU
([U4.23.01]). The characteristic length B
L is given under key word LONG_CARA of
DEFI_MATERIAU.
The constraints are the constraints of Cauchy, thus calculated on the current configuration (six
components in 3D, four in 2D).
The internal variables produced in Code_Aster are:

· V1, cumulated plastic deformation p,
· V2 with V4, the gradient following axes X, y and Z of p,
· V5, porosity F,
· V6 with V11, the tensor of elastic strain E,
· V12, the indicator of plasticity (0 if the last calculated increment is elastic, 1 if solution
figure regular, 2 if singular plastic solution).

Note:

If the user wants to possibly recover deformations in postprocessing of sound
calculation, it is necessary to trace the deformations of Green-Lagrange E, which represents a measurement of
deformations in great deformations. The traditional linearized deformations measure
deformations under the assumption of the small deformations and do not have a direction into large
deformations.

5 Bibliography

[1]
LORENTZ E.: “Laws of behavior to gradients of internal variables: construction,
variational formulation and implementation numerical ", Thèse of doctorate of the university
Paris 6, April 27, 1999.
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Code_Aster ®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables

Date:
03/02/05
Author (S):
V. CANO Key
:
R5.04.11-A Page
: 22/22

Intentionally white left page.
Handbook of Référence
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

Outline document