Code_Aster
®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. RAZAKANAIVO
Key
:
R4.04.04-A
Page
:
1/8
Manual of Reference
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
Organization (S):
EDF/AMA
Manual of Reference
R4.04 booklet: Metallurgical behavior
Document: R4.04.04
Models of metallurgical behavior
zircaloy in Code_Aster
Summary:
This document presents the models of metallurgical behavior describing structure transformations,
with the heating and the cooling, which the zircaloy (sheath of fuel pencil) between approximately 800°C undergoes
and 1000°C.
Code_Aster
®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. RAZAKANAIVO
Key
:
R4.04.04-A
Page
:
2/8
Manual of Reference
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
Count
matters
Code_Aster
®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. RAZAKANAIVO
Key
:
R4.04.04-A
Page
:
3/8
Manual of Reference
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
1 Introduction
The fuel sheaths of the nuclear jet engines with pressurized water consist of alloy of
zirconium. These alloys undergo metallurgical transformations between 800°C and 1000°C, where they
pass from a phase of compact hexagonal structure to a phase of cubic structure. In
certain cases of analysis such scenarios of accident of core per primary education loss of cooling agent (APRP),
the sheath reaches temperatures of the command 1000°C and undergoes metallurgical transformations then.
To analyze the mechanical behavior of the sheath in these situations, it is necessary of
to take into account the influence of the metallurgy on mechanics: amendment of the characteristics
mechanics, change of volume at the time of the transformations, more possibly of others
phenomena like the plasticity of transformation.
This document relates to the modeling of structure transformations of Zircaloy with
heating and with cooling and this, a scale which, while remaining “reasonable” for
metallurgist, is easily usable by the mechanic.
Metallurgy calculations in Code_Aster are done with the dedicated operator
CALC_META
, in
“postprocessing” of a thermal calculation of evolution. With the model dedicated to the transformations
austenito-ferritic of steel, one has today two models of metallurgical evolution.
choice of the model is done with the key word
RELATION
;
“ZIRC”
or
“STEEL”
. This type of modeling is
realizable within Code_Aster for the whole of the elements (
PLAN
,
AXIS
,
3D
) of
PHENOMENON
“
THERMICS
“. For the definition of the metallurgical behavior of Zircaloy the information of the word
key factor
META_ZIRC
under the control
DEFI_MATERIAU
[U4.43.01] is necessary. Lastly,
definition of the initial metallurgical state is realizable using the control
CREA_CHAMP
, under
key word factor
ETAT_INIT
of the operator
CALC_META
.
The models presented (with the heating and cooling) are formulated within the framework of the relations
of behavior with internal variables (or mémoratrices). When one carries out a sequence of
calculation thermo metal-worker-mechanics of the zircaloy, one uses the same relations of behavior
mechanics taking of account effects of the metallurgy, developed for steel:
(elasto) viscoplastic or elastoplastic, isotropic work hardening (linear or not linear) or
kinematics, taken into account or not of the phenomena of plasticity of transformation and of
restoration of metallurgical work hardening of origin.
Models established in Code_Aster to modelize the metallurgical transformations of
Zircaloy are models developed by the ECA. These models were identified on the basis of test
of dilatometry and calorimetry for alloys of sheaths (standard and new) within the framework of
collaborations EDFCEA on the behavior of the sheath in accident by loss of cool primary
(program EDGAR).
Code_Aster
®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. RAZAKANAIVO
Key
:
R4.04.04-A
Page
:
4/8
Manual of Reference
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
2
Proportion with balance.
Zircaloy has a compact hexagonal structure called phase
, stable until temperatures
about 800°C. Beyond approximately 800°C starts an allotropic transformation towards a phase
cubic
, and which is complete around 975°C.
That is to say T
deq
the initial temperature of transformation
with balance and
T
feq
the temperature of end
of transformation. The proportion of phase Z
eq
to balance is given by the equation, of type
Johnson-Mehl-Avrami, following:
[
]
<
=
<
<
-
-
-
=
<
=
eq
F
eq
eq
F
eq
D
N
eq
D
C
eq
eq
D
eq
T
T
Z
T
T
T
T
T
K
Z
T
T
Z
for
for
for
1
)
(
exp
1
0
éq
2-1
with:
T
: temperature
N
K
C
,
: parameters materials.
3
Equation of evolution to the heating
The transformation with the heating is the transformation
:
The model of evolution of phase Z
eq
to the heating is given by the differential equation (model of
Holt) following:
For
chauff
D
T
T
>
:
(
)
m
eq
Z
T
T
RT
E
With
dt
dZ
)
(
exp
-
-
=
éq 3-1
with:
)
(Z
T
eq
:
T
with balance corresponding to the proportion
Z
m
R
E
With
,
,
: parameters materials
Code_Aster
®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. RAZAKANAIVO
Key
:
R4.04.04-A
Page
:
5/8
Manual of Reference
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
4
Equation of evolution to cooling
The transformation with cooling is the transformation
The model of evolution of the phase
eq
Z
to cooling is given by the differential equation
following:
for
refr
D
T
T
<
:
)
1
(
Z
Z
K
dt
dZ
R
-
=
éq 4-1
with
(
)
)
(
exp
)
(
Z
T
T
Br
Ar
Z
T
T
K
eq
eq
R
-
+
-
-
=
NB:
The model comprises equations giving the initial temperatures of transformation of
phase according to the cooling and heating rates which today are not
operational.
5 Formulation
numerical
The integration of the differential equations is done with the explicit method of Runge-Kutta.
Note:
The discretization comprises in more one automatic cutting of the pitch of time when
C
T
°
>
5
.
5.1
Feel metallurgical evolution
One adopts the same principle as the models dedicated to steel. One considers that in a calculation of
structure, certain areas can undergo a heating while others cool. One
thus consider that there is one model of metallurgical transformations for Zircaloy which
according to the temperature considered and signs it speed of evolution thermal is described either by
model
, that is to say by the model
.
)
(
T
T
T
+
<
eq
D
T
[
]
eq
F
eq
D
T
T;
eq
F
T
>
0
)
(
>
T
T
&
0
)
(
=
T
T
&
if
eq
Z
Z
if
<
eq
Z
Z
0
)
(
<
T
T
&
Code_Aster
®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. RAZAKANAIVO
Key
:
R4.04.04-A
Page
:
6/8
Manual of Reference
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
5.2
Proportion of phase and variables internal
The programme of study of the sheath in situation of APRP (EDGAR) comprises a mechanical part. In
the mechanical model one considers that Zircaloy comprises 3 phases distinct from characteristics
different mechanics;
,
-
,
. The proportion of the three “mechanical” phases depends on
initial proportion
/
given by the equations [éq 2-1], [éq 3-1] and [éq 4-1]. Characteristics
mechanics of a multiphase point is obtained using a law of the mixtures on the characteristics of
(pseudo) phases present. This requires from the metallurgical point of view to carry out a complement
of development allowing the metallurgical exit it calculation to be able to recover 3 phases: one
phase
and a phase
who met is in the form
1 (
1
: is the phase pure alpha having
characteristic of
) is in the form
2.(
2: alpha in mixture and which will have the characteristics
mechanics of the phase
-).
The models presented previously gives the proportion of the phase
and
such as Z
=1-z
.
cold phase
is then distributed in 2 forms with proportions Z
1
and Z
2 such as:
Z
=z
1+z
2.
1>z
>0.9 0.9>z
>0
Z
1=
(
. )
.
Z
Z
-
0 9
01
Z
2=z
- Z
1
Z
1=0
Z
2=z
distribution of the proportion of phase
according to the proportion alpha
0
0,2
0,4
0,6
0,8
1
0.0,1.0,2 0,3.0,4.0,5 0,6.0,7.0,8 0,9 1
zalpha
Z
zalpha1
zalpha2
zbeta
5.2.1 Internal variables
The variables intern relation of behavior
“ZIRC”
are:
V1:
1
Z
proportion of the phase alpha1
V2:
2
Z
proportion of the phase alpha2
V3: TPG, temperature at the points of Gauss
The use of the metallurgy model dedicated to Zircaloy requires to inform the parameters of
model under the key word factor
META_ZIRC
of
DEFI_MATERIAU
.
Code_Aster
®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. RAZAKANAIVO
Key
:
R4.04.04-A
Page
:
7/8
Manual of Reference
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
6 Bibliography
[1]
RAZAKANAIVO A., DONORE AM, WAECKEL F.: Elasto-viscoplastic modeling
taking into account metallurgical transformations. Doc. of Code_Aster Reference
[R4.04.02].
[2]
WAECKEL F, RAZAKANAIVO a: metallurgical Models of behavior of steels in
Code_Aster. Doc. of Reference Code_Aster [R4.04.01].
[3]
RAZAKANAIVO A.: Introduction of a metallurgical model of transformation for Zircaloy
Report MN, 97/005, January 1997.
Code_Aster
®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. RAZAKANAIVO
Key
:
R4.04.04-A
Page
:
8/8
Manual of Reference
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
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