Code_Aster ®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. Key RAZAKANAIVO
:
R4.04.04-A Page
: 1/8
Organization (S): EDF/AMA
Handbook of Référence
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.
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
Code_Aster ®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. Key RAZAKANAIVO
:
R4.04.04-A Page
: 2/8
Count
matters
1 Introduction ............................................................................................................................................ 3
2 Proportion with balance. .......................................................................................................................... 4
3 Equation of evolution to the heating ........................................................................................................ 4
4 Equation of evolution to cooling ................................................................................................ 5
5 numerical Formulation .......................................................................................................................... 5
5.1 Feel metallurgical evolution ................................................................................................... 5
5.2 Proportion of phase and variables intern ...................................................................................... 6
5.2.1 The variables intern ............................................................................................................ 6
6 Bibliography .......................................................................................................................................... 7
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
Code_Aster ®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. Key RAZAKANAIVO
:
R4.04.04-A Page
: 3/8
1 Introduction
The fuel sheaths of the nuclear 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: modification 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 operator dedicated 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 key word RELATION; “ZIRC” or “ACIER”. This type of modeling is
realizable within Code_Aster for the whole of elements (PLAN, AXIS, 3D) of the PHENOMENE
“THERMIQUE”. For the definition of the metallurgical behavior of Zircaloy the information of the word
key factor META_ZIRC under command DEFI_MATERIAU [U4.43.01] is necessary. Lastly,
definition of the initial metallurgical state is realizable using command CREA_CHAMP, under
key word factor ETAT_INIT of 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 model 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).
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
Code_Aster ®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. Key RAZAKANAIVO
:
R4.04.04-A Page
: 4/8
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 eq
eq
D the initial temperature of transformation to balance and Tf
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
Z
= 0
for
T < eq
T
D
eq
N
eq
Z
= 1 - exp
- [KC T
(- T)
for
T
T
T
D
]
eq < < eq
D
F
éq
2-1
eq
eq
Z
= 1
for
T < T
F
with:
T: temperature
K
N
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
T > Td
:
dZ
E
= Aexp-
(T - T ())m
eq Z
éq 3-1
dt
RT
with:
T (Z)
eq
: T with balance corresponding to proportion Z
E
With,
, m: parameters materials
R
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
Code_Aster ®
Version
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Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. Key RAZAKANAIVO
:
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4
Equation of evolution to cooling
The transformation with cooling is the transformation
The model of evolution of the phase eq
Z with cooling is given by the differential equation
following:
for
refr
T < Td:
dZ = K Z 1 (- Z)
R
éq 4-1
dt
with
K = - T - T (Z) exp
R
eq
(Ar+BrT - T (Z)
eq
)
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 step of time when
T
> 5 C
°.
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 zones 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, is by the model.
T (T + T
) < eq
T
[eq eq
T T
;
eq
T
>
D
F
]
D
F
T& (T) > 0
T& (T) = 0 if Z Z
if Z <Z
eq
eq
T& (T) < 0
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
Code_Aster ®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. Key RAZAKANAIVO
:
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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 which met is in the form 1 (1: is the phase pure alpha having
characteristic of) is in 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 the proportions z1 and z2 such as:
z=z1+z2.
1>z>0.9 0.9>z>0
(Z -.
0 9)
z1=0
z1=
Z
.
01
z2=z
z2=z-z1
distribution of the proportion of phase
according to the proportion alpha
1
0,8
zalpha1
0,6
Z
zalpha2
0,4
zbeta
0,2
0
0.0,1.0,2 0,3.0,4.0,5 0,6.0,7.0,8 0,9 1
zalpha
5.2.1 Internal variables
The internal variables of the relation of behavior “ZIRC” are:
V1: z1 proportion of the phase alpha1
V2: Z 2 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.
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
Code_Aster ®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. Key RAZAKANAIVO
:
R4.04.04-A Page
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6 Bibliography
[1]
RAZAKANAIVO A., DONORE AM, WAECKEL F.: Elasto-viscoplastic modeling
taking into account metallurgical transformations. Doc. of Référence Code_Aster
[R4.04.02].
[2]
WAECKEL F, metallurgical RAZAKANAIVO a: Modèles of behavior of steels in
Code_Aster. Doc. of Référence Code_Aster [R4.04.01].
[3]
RAZAKANAIVO A.: Introduction of a metallurgical model of transformation for Zircaloy
Report MN, 97/005, January 1997.
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A
Code_Aster ®
Version
5.0
Titrate:
Models of metallurgical behavior of Zircaloy
Date:
20/08/02
Author (S):
A. Key RAZAKANAIVO
:
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: 8/8
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R4.04 booklet: Metallurgical behavior
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