Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 1/12
Organization (S): EDF-R & D/MMC
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
Document: R5.03.07
Model of Rousselier for the ductile rupture
Summary
The model of Rousselier describes the damage due to the plastic growth of cavities in a metal. It
allows to model cracking and the ductile rupture. The relation of behavior is elastoplastic or
viscoplastic with isotropic work hardening. It allows the changes of plastic volume and is written in
small deformations. The writing in great deformations with a formulation of Simo and Miehe modified, in
the elastoplastic case only, is described in [R5.03.06].
This model is available in command STAT_NON_LINE via the key word RELATION =
“ROUSS_PR” or “ROUSS_VISC” under the key word factor COMP_INCR and with the key word DEFORMATION =
“PETIT_REAC”.
This model is established for modelings three-dimensional (3D), axisymmetric (AXIS), in constraints
plane and in plane deformations (C_PLAN, D_PLAN).
One presents the writing and the digital processing of this model.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 2/12
Count
matters
1 Introduction ............................................................................................................................................ 3
2 Notations ................................................................................................................................................ 4
3 Model of Rousselier ............................................................................................................................ 5
3.1 Derivation of the equations of the model ............................................................................................... 5
3.2 Equations of the model ....................................................................................................................... 7
4 numerical Formulation .......................................................................................................................... 8
4.1 Key words, given material and variables intern ......................................................................... 8
4.2 Expression of the model discretized ..................................................................................................... 8
4.3 Resolution of the nonlinear scalar equation ............................................................................... 10
4.4 Form of the tangent matrix of the behavior ................................................................... 10
5 Bibliography ........................................................................................................................................ 12
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 3/12
1 Introduction
The mechanisms at the origin of the ductile rupture of metals are associated the development of
cavities within material. Three phases are generally distinguished:
· germination: it is about starting or nucleation of the cavities, in sites which correspond
preferentially with the particles of second phase present in material,
· growth: it is the phase which corresponds to the development itself of the cavities,
controlled primarily by the plastic flow of the metal matrix which surrounds these
cavities,
· coalescence: it is the phase which corresponds to the localization of the deformation between the cavities
to create macroscopic fissures.
The model of Rousselier [bib1], [bib2], [bib3] presented here is based on assumptions
microstructural which introduces a microstructure made up of cavities and of a matrix of which them
elastic strain negligible are compared with the plastic deformations. In this case, and in
the absence of nucleation of new cavities, porosity F, definite like the relationship between volume
cavity C
V and total volume V of representative elementary volume, is directly connected to
macroscopic plastic deformation by:
C
1 - F
0
0
V
=
with F =
F & = (1 - F)
p
tr &
éq
1-1
1 - F
V
where f0 indicates initial porosity, and are respectively the density in
O
configurations initial and current (one takes in the continuation = 1) and p
O
& the rate of deformation
plastic of total volume V.
The construction of the model rests on a thermodynamic and phenomenologic analysis which brings
to write the plastic potential F in the following form:
(
F, p, F) = + D F exp m - R (p) éq
1-2
eq
1
1
1
where =/is the constraint of Kirchhoff, is the constraint of Cauchy, R isotropic work hardening
function of the cumulated plastic deformation p, and D of the parameters of material. The presence
1
1
in the plastic potential of the hydrostatic constraint authorizes the changes of volume
m
plastic.
In the event of nucleation of new cavities, one considers that the voluminal fraction created is
proportional to the cumulated plastic deformation. It is thus enough to replace F by F + A p in
N
equations of the model. A is a parameter of material. The equation [éq 1-1] is not modified.
N
In the viscoplastic case, one writes the viscoplastic potential vp
F like a function of the potential
plastic F:
vp
F = (,
F p, F)
éq 1-3
One will consider only the particular case such as:
m
F
p & =
= &
éq 1-4
0 HS
F
0
who is reduced to a function power (law of the Norton type) when two parameters of material
& and are very large.
0
0
Thereafter, one presents the relations of behavior of the model of Rousselier and his integration
numerical.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 4/12
2 Notations
One will note by:
Id
tensor second-order identity
II
tensor identity of the fourth command
tr A
trace second-order tensor A
~
~
1
With
With = A - (tr A)
deviatoric part of tensor A defined by
Id
3
tr A
With
With =
hydrostatic part of tensor A defined by
m
m
3
3
With
~ ~
eq
equivalent value of von Mises defined by Aeq =
:
WITH A
2
T
:
doubly contracted product: A: B = A B = tr (AB)
ij ij
I, J
tensorial product: (A B)
=
ijkl
ij
With kl
B
, µ, E, K
moduli of the isotropic elasticity
2
p&
~
~
speed of equivalent plastic deformation
p
p
p =
&
&
: &
3
In addition, within the framework of a discretization in time, all the quantities Q evaluated at the moment
precedent are subscripted by -, quantities evaluated at the moment T = T - + T
are not subscripted and
the increments are indicated par. One has as follows:
Q = Q + Q
The numerical resolution is carried out by one - method, with 0 1. For all the quantities,
one defines:
Q = Q + Q
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 5/12
3
Model of Rousselier
We now describe the derivation of the equations of the model of Rousselier presented in
introduction.
3.1
Derivation of the equations of the model
It is supposed that the specific free energy breaks up into three parts: a hyperelastic part
who depends only on the elastic strain, a part related to the mechanism of work hardening and one
part related to the damage:
(E
, p, F) = E (E
) + p (p)
F
+ (F)
éq
3.1-1
The inequality of Clausius-Duhem is written (one does not consider the thermal part):
: & - & 0
éq 3.1-2
expression in which
E
p
& = & + & the rate of deformation represents.
Dissipation is still written:
E
p
-
: & +: & -
p & -
F & 0 éq
3.1-3
E
p
F
The second principle of thermodynamics then requires the following expression for the relation
elastic stress-strain:
=
éq 3.1-4
E
One defines the thermodynamic forces associated with the elastic strain, with the deformation
figure cumulated and with porosity in accordance with the framework of generalized standard materials:
(E
) =
éq 3.1-5
E
(
p) =
With
éq 3.1-6
p
(
F) =
B
éq 3.1-7
F
It remains then for dissipation:
: p
& - A p & -
B F & 0
éq
3.1-8
The principle of maximum dissipation applied starting from the viscoplastic potential Fvp (, A, B) allows
to deduce the laws of evolution from them from the plastic deformation, the plastic deformation cumulated and of
porosity, is:
p
vp
F
& =
éq 3.1-9
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 6/12
F
vp
p & = -
éq 3.1-10
With
F
vp
F & = -
éq 3.1-11
B
It is supposed that Fvp (, A, B) is a function of the plastic potential F (, A, B) and that this last
break up into two terms depending respectively on the second invariant on coupled to A and on
first invariant of coupled to b:
Fvp = (F) = (F
+
éq
3.1-12
VM (
, A
eq
) Fm (, B
m
)
By assumption, the first term breaks up in an additive way like the potential of von Mises:
F
= -
-
= -
éq
3.1-13
VM (
, A
eq
)
With
eq
(p) R
R (p)
0
eq
Not to obtain a commonplace result, the decomposition of the second term must be multiplicative:
F
=
éq
3.1-14
m (
, B
m
) G (m) H (B)
Taking into account the equation [éq 1-1], the laws of evolution for
p
tr & and F & lead to the equality:
g' (
- 1 B
m)
H ((F)
=
éq
3.1-15
G (
1
- B
m)
F H ((F)
The two members of this equation are functions of the two independent variables and F,
m
thus they is equal to a constant of dimension the reverse of a constraint, it is the parameter of
material 1/. The parameter without dimension D appears in the integration of G/G:
1
1
G ()
= D exp m
éq
3.1-16
m
1
1
1
The function (
B F) and the function reverses F = H (B are unknown. The simplest choice and more
1
)
naturalness is to take H H, which gives:
1
H (B) H
éq 3.1-17
1 (B) = F
(B)
F
D
1
H
=
= -
F (1 - F)
éq
3.1-18
dB
1
The plastic potential is written finally:
F = + D F exp m - R (p) éq
3.1-19
eq
1
1
1
The law of evolution for p & gives:
D (F)
p & =
= V (F)
éq 3.1-20
dF
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 7/12
The function V (F) defines the viscosity of material. One will consider only the particular case such as:
m
F
V (F) = &
éq
3.1-21
0 HS
0
who is reduced to a function power (law of the Norton type) when two parameters of material
& and are very large. Conversely one a:
0
0
F - S (p&) = 0
éq 3.1-22
1
1
m
p&
S (p&) =
-
HS
éq
3.1-23
0
&0
In the case of plasticity independent of time, the preceding equation becomes F = 0 (criterion or
threshold of plasticity) and p & is given by the equation of consistency F & = 0 if F = 0 and p & = 0 if F < 0.
The equations of the model are now completely defined, in the case without nucleation of
new cavities. In the event of nucleation of new cavities, one considers that the voluminal fraction
created is proportional to the cumulated plastic deformation. It is thus enough to replace F by
F + A p in the equations of the model. A is a parameter of material. The equation [éq 1-1] is not
N
N
not modified.
3.2
Equations of the model
One summarizes the equations of the model deduced from the thermodynamic and phenomenologic analysis which
precede:
1
m
p
m
-
&
= + D F + A p
- R p - HS
=
éq 3.2-1
vp
eq
1
1 (
N
) exp
()
1
0
0
1
&0
=
= [(Id Id) + 2µII] E
:
éq
3.2-2
1 - F - A p
N
=
éq 3.2-3
1 - f0
~
~
~ p
3
3
& = p&
= p
éq 3.2-4
&
2
2
eq
eq
p
tr & = p&D F A p exp m
éq
3.2-5
1 (
+ N)
1
F & = A (1
éq 3.2-6
1
) p
F tr &
with A = 1, this parameter being introduced only for numerical reasons.
1
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 8/12
4 Formulation
numerical
4.1
Key words, given internal material and variables
For the foreseeable applications, the model was established under two distinct key words:
`ROUSSELIER_PR `for the plastic model with nucleation of cavities or `ROUSSELIER_VISC `for
the viscoplastic model without nucleation. That makes it possible to avoid useless numerical calculations.
corresponding simplified equations are obtained starting from the general equations while posing
respectively = 0 or A = 0.
0
N
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 ([U4.43.01]). Parameters of the viscoplastic model (,
0
& and m) are provided
0
by the key word `ROUSSELIER_VISC `.
The internal variables produced in Code_Aster are:
·
V1, cumulated plastic deformation p,
·
V2, porosity F,
·
V3 with V8, the tensor of elastic strain E
,
·
V9, the indicator of plasticity (0 if the last calculated increment is elastic, 1 if solution
figure regular, 2 if singular plastic solution).
We now present the numerical integration of the law of behavior and give
the form of the tangent matrix (options FULL_MECA and RIGI_MECA_TANG).
4.2
Expression of the discretized model
The numerical resolution is carried out by one - method, with 0 1. For all the quantities
Q, one defines:
Q = Q + Q
Q = Q + Q
The system of equations discretized is:
~
~e
= 2µ = 2µ (~
~ p
-) éq
4.2-1
E
= Ktr
= K tr
- tr
éq
4.2-2
m
(
p)
~
p
3
~
= p
éq 4.2-3
2 eq
p
tr = Pd F
With p exp m
éq
4.2-4
1 (
+ N)
1
F
= A
éq 4.2-5
1 (1 -
)
p
F
tr
1
m
p
m
-
=
+ D F + A p
- R p - HS
= éq 4.2-6
vp
eq
1
1 (
N
) exp
(
)
1
0
0
&
T
1
0
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 9/12
This system is reduced to the solution of only one equation scalar for the unknown factor F
, knowing
, T
and quantities -
Q. It is noted that does not intervene in the algorithm, on the other hand it
will intervene in the calculation of the coherent tangent matrix. One calculates successively:
F
=
-
+ K tr
éq
4.2-7
m
m
-
A1 (1 -
F)
p
is the positive root of the quadratic equation:
With
F
p
F
With p
p
éq
4.2-8
N
() 2 + (+
-
N
)
1
-
=
With -
F
D
1 (1
) exp
1
(m 1) 0
/
3
µ
p
~ = 1
-
- [
~
~
+ 2
µ
éq
4.2-9
~ - +
µ ~
2
] (
)
eq
= ~-
+
µ ~
2
- 3
µ
éq
4.2-10
eq
[
]
p
eq
The scalar equation for F
is the equation [éq 4.2-6] = 0.
vp
Notice 1:
Like F
is very weak in most of the structure, it would be preferable
to use
p
like principal unknown factor. But in this case it is not possible
to bring back to a scalar equation, which makes more difficult the use of a method of the type
Newton. It is also one of the reasons why the equations [éq 1-1], [éq 3.2-6] and
[éq 4.2-5] were not modified by the introduction of the nucleation of the cavities.
Notice 2:
The equation [éq 3.2-6] can be integrated exactly:
1
1
p
- F
tr =
ln
0
A1 1 - F
from where:
1
p
1 - -
F
tr =
ln
With
1
1 - F
As numerical parameter A can be modified in a discontinuous way, the derived form
1
[éq 4.2-5] was preserved, including in the calculation of the coherent tangent matrix. If
the use of parameter A was to be abandoned in a later version, it would be necessary
1
to consider the use of the integrated form.
Notice 3:
The integrated form = 0 is used, including in plasticity instead of the relation of
vp
consistency F & = 0 which gives p &. The coherent tangent matrix is calculated with this
form integrated.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 10/12
4.3
Resolution of the nonlinear scalar equation
The resolution of the equation
(F
be carried out by an algorithm of Newton on controlled terminals
vp
) = 0
in routine LCROUS. (F
and its derivative compared to F
are calculated in the routine
vp
)
RSLPHI called by LCROUS. The initial values of the terminals are:
· limit lower: F = 0 since
(it was checked as a preliminary that the branch
vp (0) < 0
1
rubber band (negative threshold) is not solution),
· limit higher: F
such as
sought by dichotomy between 0 and
-
1 - F
vp (0) > 0
2
1
-
- F
(first value for this search:
).
2
The algorithm of Newton begins with the value F = 0. Whatever the value found for F
one
thus note for the continuation that the function (F
and its derivative compared to F
are at least
vp
)
1
-
- F
calculated for F = 0 and
.
2
The developments carried out to improve convergence and the robustness of the algorithm are
described in [bib5].
4.4
Form of the tangent matrix of the behavior
One gives the form of the tangent matrix here (option FULL_MECA during iterations of
Newton, option RIGI_MECA_TANG for the first iteration).
For option RIGI_MECA_TANG, the tangent operator is the same one as that which connects E
with in
[éq 3.2-2].
For option FULL_MECA, the tangent matrix is obtained by linearizing the system of equations which
governs the law of behavior: [éq 4.2-1] with [éq 4.2-6]. It is thus about a coherent tangent matrix.
To simplify the expressions, one notes in this paragraph [§4.5]: Q for
Q, quantities being
all expressed at the moment T = T - + T
. The coherent tangent matrix is:
has - has
has
has
y
1
3
~ ~
~
5
1
= II + Id has
has
has
y
3
Id +
2
+ +
Id
4
+
5 ~
4
-
1 Id +
3
3
3K
K
éq 4.4-1
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 11/12
This operator is calculated in routine RSLJPL. The coefficients are calculated as follows:
= 3K has + y K (Z + Z p
)
éq
4.4-2
1
1
eq
7
2
= µ has (y + y)
éq 4.4-3
2
1
3
1
2µ
has
eq
=
éq 4.4-4
3
z5
= 3µy X has
éq 4.4-5
4
2 2
= 3µy has
éq 4.4-6
5
1
1
= 3µK p has
- has
éq 4.4-7
6
2 eq
1
3Kz Z (F + A p)
6 1
N
y = -
éq 4.4-8
1
X
1 eq
3µ
y = -
éq 4.4-9
2
2
X Z
1 5 eq
3Kz Z A p
6 1
N
y = -
éq 4.4-10
3
X
1 eq
With Z
Z
y = 1 8 +
9
1
éq 4.4-11
4
Z
Z + Z p
1
7
2
A to Z
Z has
1 2 8
9 6
y =
-
éq
4.4-12
5
Z
(Z + Z p
)
1
eq
7
2
Z = 1+ A
Pd (F + A p) exp m
éq
4.4-13
1
1
1
N
1
Z = µ
3 + R
éq 4.4-14
2
vp
Z = K (F + A p) Z - A 1
(- F) éq
4.4-15
3
N
1
1
1
Z = R p
-
éq 4.4-16
4
vp
eq
Z = + µ
3
p
éq 4.4-17
5
eq
Z =
exp m
D
éq 4.4-18
6
1
1
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Code_Aster ®
Version
6.4
Titrate:
Model of Rousselier for the ductile rupture
Date:
10/09/04
Author (S):
G. ROUSSELIER, R. MASSON, G. Key BARBER
:
R5.03.07-A Page
: 12/12
Z = Z (F + A p)
éq 4.4-19
7
6
1
N
1 - F
Z =
éq 4.4-20
8
1 - F - A p
N
With
Z
N
=
éq 4.4-21
9
1 - F - A p
N
X = Z Z (Z + Z p
) + Z Z - X
éq
4.4-22
1
3 6
7
2
1 2
1
3
X = - Z Z p
(Z + Z) - Z Z + X p
éq
4.4-23
2
3 6
4
7
1 4
1
3
2
X = A Z Z
éq 4.4-24
3
N 1 6
1
Dr. (p)
1 dS (p
/T
)
R =
+
éq
4.4-25
vp
dp
T
p
d&
For the plastic model with nucleation of cavities `ROUSSELIER_PR `and for the model
viscoplastic without nucleation `ROUSSELIER_VISC `, the corresponding simplified equations are
obtained starting from the equations above by posing R respectively = Dr. (p)/dp and A = 0.
vp
N
5 Bibliography
[1]
ROUSSELIER G.: “Finite constitutive deformation relations including ductile fracture
ramming ", in Three-Dimensional Constitutive Relations and Ductile Fracture, ED. Nemat-
Nasser, North Holland publishing company, pp. 331-355, 1981.
[2]
ROUSSELIER G.: “The Rousselier model for porous metal plasticity and ductile fracture”, in
Handbook off Materials Behavior Models, ED. J. Lemaitre, Academic Press, pp. 436-445,
2001.
[3]
ROUSSELIER G.: “Dissipation in porous metal plasticity and ductile fracture”, J. Mech. Phys.
Solids, vol. 49, pp. 1727-1746, 2001.
[4]
BARBER G.: “Model of Rousselier in Code_Aster: new implementation”, notes
EDF R & D HT-2C/98/007/A, 1998.
[5]
MASSON R., ROUSSELIER G., BONNAMY Mr.: “Improvement of convergence and evolution
implementation of the model of Rousselier in Code_Aster ", note EDF R & D
HT-26/01/037/A, 2002.
[6]
MIALON P.
: “Elements of analysis and numerical resolution of the relations of
elastoplasticity ", EDF, bulletin of DER, data-processing series C mathematical, 3, pp. 57-89,
1986.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/04/002/A
Outline document