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Organization (S): EDF/MTI/MN
Handbook of Référence
R4.04 booklet: Metallurgical behavior
Document: R4.04.01
Models of metallurgical behavior
steels in Code_Aster
Summary:
This document presents the models of metallurgical behavior at the heating and cooling
allowing to describe structure transformations of steels at the time of cycles thermal exceeding of
temperatures about 800°C.
For the two types of transformations (with the heating and cooling) a detailed description of
models available is made and of the methods of identification are given.
Lastly, one presents the model of calculation of hardness associated with the metallurgy.
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Models of metallurgical behavior of steels
Date:
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Contents
1 Introduction ............................................................................................................................................ 3
2 Model of metallurgical behavior to cooling ................................................................. 4
2.1 Introduction ...................................................................................................................................... 4
2.2 Assumptions ...................................................................................................................................... 5
2.3 Choice of the variables of state ............................................................................................................... 6
3 Identification and implementation of the model to cooling ............................................................. 9
3.1 Principle ............................................................................................................................................ 9
3.2 Integration of the experimental data to the model ....................................................................... 9
3.2.1 Principle .................................................................................................................................. 9
3.2.2 Rules of interpretation of diagrams TRC ..................................................................... 10
3.2.3 Effect of the austenitic size of grain on the kinetics of the transformations with
cooling. ................................................................................................................... 12
3.2.4 Data entry of diagrams TRC ............................................................................................... 13
3.3 Evaluation of the function of evolution starting from the experimental data .................................. 14
3.3.1 Evaluation of the function of evolution for the experimental stories .............................. 14
3.3.2 Calculation of the advance of the transformations for an unspecified state ................................ 14
4 Model of metallurgical behavior to the heating ........................................................................ 17
4.1 Assumptions .................................................................................................................................... 17
4.2 Form model selected ............................................................................................................... 18
4.3 Integration of the equation of evolution .............................................................................................. 19
4.4 Evolution of the austenitic size of grain to the heating .............................................................. 19
4.4.1 Digital processing .......................................................................................................... 20
4.5 Feel metallurgical evolution .................................................................................................. 20
5 Identification of the model to the heating ................................................................................................. 21
5.1 Determination of the function Zeq (T) ............................................................................................. 21
5.2 Determination of function TAUX (T) ......................................................................................... 21
5.2.1 Identification of TAUX_3 starting from AC'3 ............................................................................ 22
5.2.2 Identification of TAUX_1 starting from AC'1 ............................................................................ 22
6 Model of calculation of hardness .................................................................................................................. 23
7 Bibliography ........................................................................................................................................ 24
Appendix 1 ................................................................................................................................................. 25
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1 Introduction
Operations of assembly and thermomechanical processing which the components undergo
metal of power stations REP, can generate, in the materials which constitute them, of the fields
constraints, known as residual, which exist before any loading of service. Transformations
metallurgical and mainly structure transformations are an important cause of these
stress fields because they modify the behavior (by modifying the characteristics
physics) and generates mechanical thermo stresses within the materials which undergo them
(latent heats, deformations due to the differences in density of the various phases
metallurgical).
These structure transformations are due essentially to the succession of a heating (often with
beyond 800 °C) and of a more or less fast cooling of the parts during their manufacture. These
thermal “cycles” can desired (case of the heat treatments) or “be undergone” (case of welding).
In all the cases, they are very variable of a point to another of the part.
This document relates to the modeling of these structure transformations to the heating and with
cooling for low alloy steels and this, on a scale which, while remaining “reasonable”
for the metallurgist, that is to say easily usable by the mechanic.
This type of modeling is realizable within Code_Aster for the whole of elements (PLAN,
AXIS, 3D) of PHENOMENE “THERMIQUE” by the use of operator CALC_META in “post-
processing “of a thermal calculation of evolution. The relation of behavior dedicated to steel, by
difference with that dedicated to the zircaloys, to use under key word COMP_INCR of the operator is
relation “ACIER”. For the definition of the metallurgical behavior of steel the information of
order DEFI_TRC and key word factor META_ACIER under command DEFI_MATERIAU is
necessary. Lastly, the definition of the initial metallurgical state is realizable using the command
CREA_CHAMP, under the key word factor ETAT_INIT of operator CALC_META. The metallurgy calculation
is necessary to the realization of mechanical calculations which take account of the consequences
mechanics of these metallurgical phenomena [R4.04.02]
The models presented (with the heating and cooling) are formulated within the framework of the relations
of behavior with internal variables (or mémoratrices), and authorize a simple identification and
rapid based on the experimental diagrams (diagrams TRC of Transformations in
Continuous cooling). The choice of the variables and the forms of the laws of evolution selected are
given and the description of the implementation of the models (method of identification) is also
presented.
Lastly, one presents the model of calculation of hardness which can if necessary come to supplement calculations
metallurgical.
Note Bucket:
· Basic metallurgical notions necessary to comprehension of the general problem
and of the adopted step are gathered in [bib1] and [bib2] where one will find
also a bibliographical study of the problem.
· This document is extracted from [bib3] and [bib4] where one makes a more detailed presentation of
models and of some elements of validation. More complete elements of validation
can also be found in [bib5] for the model of cooling and in [bib14]
for the model of hardness.
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2
Metallurgical model of behavior to cooling
2.1 Introduction
On the basis of test of dilatometry [Figure 2.1-a], only knowledge, at a given moment,
temperature of an undergoing steel of structure transformations does not make it possible to know its state
of deformation. On the other hand, the behavior of such a steel seems to be able to be described within the framework
models of behavior to variables mémoratrices or interns [bib6]. Indeed, if one introduces:
·
Z = {Z I =
} p
I;
,
1
the p-uplet of the proportions of the possible metallurgical components
present in a point M and at one moment T given (here, Z1, Z2, Z3, Z4 will be the proportions
of ferrite, pearlite, bainite and martensite and the proportion of austenite in M will be equal to:
1 - Z
(
)
1 + Z2 + Z3 + Z4);
·
HT
HT
(T) = (T - T) and (T) = (T - T) + (T) thermal deformations
austenite and phases ferritic, perlitic, bainitic and martensitic; while noting:
-
the thermal dilation coefficient average of austenite;
-
T the temperature of reference to which one considers HT
null;
-
the thermal dilation coefficient average presumedly identical for ferrite,
pearlite, the bainite and martensite;
-
deformation, at the temperature T, of the phases ferritic, perlitic, bainitic and
martensitic compared to austenite (by taking the latter like the phase of
reference);
· if it is considered, moreover, that the deformation of a multiphase mixture can be obtained with
to leave the deformations of each phase by a linear law of mixture, one can then describe
evolution of the state of deformation during a dilatometric test by:
I = 4
I
= 4
HT (Z, T)
HT
HT
= 1 - Z
I
(T) Zi (T)
+
I = 1
I = 1
éq 2.1-1
I = 4
I
= 4
= 1 - Z
I
[
(T - T)] + Zi
[(T - T) +].
I = 1
I =1
The problem lies then in the determination of Z or, more precisely and within the framework of
simple materials with variables mémoratrices, in the determination of the function of evolution F such
that: !
Z = F (T, Z,)
… .
To give an account of an effect the speed of cooling on the evolution of the transformations
structural, we propose, within the framework of simple materials with variables mémoratrices, one
modeling of the metallurgical behavior of steels to the cooling which includes, a priori!
T among
its variables of state.
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Austenitization
1
2
3
T
1: speed of cooling = 1 °C/min
2: speed of cooling = 10 °C/S
3: speed of cooling = 100 °C/S
Appear 2.1-Error! Argument of unknown switch. : Diagrammatic curves of dilatometry
2.2 Assumptions
H1:
A steel likely to undergo structure transformations is a simple material with
variables mémoratrices among which one can choose characterizing quadruplet Z
metallurgical structure in a given point and at one moment.
One thus models structure transformations on a scale where the material point can
to be multiphase. This scale of modeling which can appear métallurgiquement
coarse is in conformity with the concept of material point used in mechanics of the mediums
continuous and the test-tube of dilatometry, presumedly homogeneous, is representative.
H2:
Supplemented diagrams TRC of the martensitic kinetics of transformation of
Koistinen-Marburger [bib7] completely characterize the metallurgical behavior of one
steel austenitized during a continuous cooling.
This assumption results directly from the metallurgical practice and specifies the first of
objectives to be fixed at the model: to be compatible with the whole of the experimental data
relating to the metallurgical behavior which accompanies cooling by steels
austenitized. In addition, this assumption also generates a “natural” choice and
restrictions as for the variables to be introduced into the model.
H3:
The transformations ferritic, perlitic and (especially) bainitic are impossible in on this side
martensitic initial temperature of transformation ms.
This assumption, in conformity with the representation of diagrams TRC, makes it possible to uncouple
transformations by diffusion of the martensitic transformation.
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2.3
Choice of the variables of state
Pilot variables of state
Into thermomechanical of the continuous mediums, the variables of state control are generally
temperature and the state of constraints or deformations. However, because of the assumption H2,
temperature is only the variable pilot retained. Indeed, the influence of the state of stresses on
structure transformations does not appear in diagrams TRC. Moreover, there does not exist (except
an effect of the type Châtelier) of ideal model even if experimental data relative to
this influence in isothermal conditions were obtained for certain steels [bib8].
Variables of state mémoratrices
The first variable mémoratrice to be introduced is quadruplet Z characterizing the structure
metallurgical and to which knowledge is enough, a priori, to describe from a mechanical point of view a test
dilatometric [§2.2].
In addition to the temperature T, its derivative!
T and the state of stresses, austenitic size of grain D and
percentage of carbon C of austenite changing also influence the behavior
metallurgical of steels to cooling. However, always because of the assumption H2, one
chooses not to introduce C like variable mémoratrice. Indeed, diffusion of carbon
does not appear explicitly on diagrams TRC, although it is implicitly taken into account,
at least partially, in the concept even of component metallurgical. In addition Giusti showed
that if the taking into account of C were theoretically possible, it led to equations
of evolution coupled between C and Z whose experimental identification “seems very difficult, for not
to say impossible " [bib9]. Nevertheless, an effect of the percentage of carbon on the decomposition of austenite
with cooling appears indirectly on diagrams TRC. It is the phenomenon of
stabilization of the austenite which results in a decrease of the temperature of transformation
martensitic ms [Figure 2.3-a].
Contrary to the percentage of carbon, the austenitic size of grain D appears on the diagrams
TRC which relate to conditions of austenitization to which correspond a value of D.
We thus choose to introduce D like variable mémoratrice. However, size of grain
austenitic, which results from the thermal history undergone with the heating does not evolve/move more with cooling and
D intervenes only as a parameter in the model of behavior to cooling.
In addition, the martensitic temperature of transformation ms, which depends on the thermo history
metallurgical undergone, intervenes in the law of Koistinen-Marburger adopted on the assumption H2 for
to describe the martensitic transformation. One thus chooses to introduce ms like variable mémoratrice.
The character memorator of the variables mémoratrices introduced here in addition to Z appears clearly:
D characterizes the thermal history undergone at the time of the passage in austenitic phase and ms connects
decomposition of austenite in the conditions of its transformation into martensite.
Relation “ACIER” of operator CALC_META thus comprises 7 internal variables:
V1: Z1, proportion of ferrite,
V2: Z2, proportion of the pearlite,
V3: Z3, proportion of bainite,
V4: Z4, proportion of martensite,
V5: D, austenitic size of grain,
V6: Ms martensitic temperature of transformation,
V7: temperature at the points of Gauss.
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Appear 2.3-Error! Argument of unknown switch. : Example of diagram TRC
It is also necessary to model the whole of the phenomena brought into play at the time of one
operation of welding to introduce other variables mémoratrices such as the tensors of
anelastic deformations which can correspond to the plastic deformations, of plasticity of
transformation or of viscosity. But, in accordance with the assumption H2, one considers that these variables
do not intervene in the functions of evolutions of Z and ms.
Lastly, the following assumptions make it possible to simplify and specify more the general form of
model.
H4:
T intervenes only in the relation of behavior expressing the current vector of
.
heat Q; its temporal derivative first T is not a variable of state and the relation of
behavior expressing the current vector of heat is the Fourier analysis:
Q = - (T, Z, D) T.
H5:
I = 3
A diagram TRC makes it possible to identify an empirical relation between ms, D and Zi:
I = 1
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=
+
(
I
3
Ms Z, Z, Z; D
1
2
3
) = Ms0 (D) + AM (D) Z - Zs (D
I
).
éq 2.3-1
I = 1
The H5 assumption means that the martensitic initial temperature of transformation is
constant (for a size of grain given) and equal to Ms0 as long as the proportion of austenite
transform is lower than a threshold Z S and than its variation is a linear function of
quantity of transformed austenite. This assumption seems checked relatively well
in experiments [2.3 - has]. It makes it possible to exclude ms from all the relations of
behavior other than that expressing Z and Z4.
With Z = {Z, Z, Z which one will distinguish well from Z = {Z I
I;
=,
1}
p defined in the § 2.1.
1
2
3}
Finally, and taking into account the assumptions H2 and H3 the relations defining the model are written
thus:
+
.
T - Ms
!(
Z T) = F (T!T, Z ms; D)
[
]
= F T, T, Z; D
with Z = {Z, Z, Z
éq 2.3-2
1
2
3}
T - Ms
I = 3
+
Z
4 (T,
Z ms; D) = 1 -
Z
-
-
éq 2.3-3
{1 ex (
p (D) [ms T
I
])}
I = 1
and
+
I
= 3
(
Ms T) = Ms0 (D) + A (D).
Z
- Zs (D
M
I
) éq
2.3-4
I
= 1
where: is a characteristic of the material (°C-1) (possibly function of D);
and [X] + the positive part of X indicates.
Lastly, as it seems difficult to propose a simple form of dependence of the model with respect to
these variables, one chose not to impose of form particular to the functions of evolution fi [bib2].
The step to calculate speeds of evolution of the metallurgical variables uses then
techniques of interpolation and rests on the fact that any thermometallurgic history
in experiments known (dilatometric test for example) is a particular solution of
the differential equation of evolution [éq 2.3-2].
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3
Identification and implementation of the model with
cooling
3.1 Principle
Identification of the model and the use of the experimental data which the diagrams constitute
TRC to determine the value taken by the function F in a metallurgical thermo state (T, T, Z; D)
given are founded on the following observation and the assumption:
· the metallurgical thermo stories being reproduced on a diagram TRC are all of the solutions
particular of the differential equation [éq 2.3-2]. They thus make it possible to calculate in each
thermodynamic state met in experiments and present in a diagram TRC
value taken by the function F.
·
function
F is regular; i.e. if two points E and E
K
J are close
(Ek = {T (tK), T!(tK), (ztK);D (tK)}), their speeds of evolution in Z are also
neighbors is:
T - Ms +
T - Ms +
E
E F (E) [
] =!(zE) F (E) [
] =!(zE
K
J
K
K
J
J).
T - Ms
T - Ms
One determines then speeds of structure transformations of an unspecified state by
interpolation among all the “couples” (E, F (E
K
K) defined by diagrams TRC.
3.2
Integration of the experimental data to the model
3.2.1 Principle
In general, a diagram TRC defines in a reference mark [(
ln T) - T] structure transformations
associated a series of thermal stories traced on this diagram [Figure 2.3-a]. The integration of
experimental data then consists in recording for each history of these diagrams the values
successive of T, T!, Z so that for any temperature T the model knows
values taken by the function F in (T, T!(T), (
Z T)). In order to be able, starting from a reduced number of
numerical data, to reconstitute the thermometallurgic evolutions continuously, one formulates
some assumptions on the thermal evolutions and the metallurgical behavior of steels.
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3.2.2 Rules of interpretation of diagrams TRC
Thermal evolutions
To define the thermo stories metallurgical present in a diagram TRC it is necessary to characterize
their thermal evolutions. One can notice that, in a reference mark [1 (
N T) - T] and for
temperatures lower than 820 °C, the thermal stories of diagrams TRC can, with one
enough good approximation, to result from/to each other by a horizontal adjustment
[Figure 2.3-a]. It is thus possible to define a thermal history T I (T) starting from the data of one
curve controls Tp T
() and of the moment (in second) for which this history crosses the isotherm 820 °C
by:
T I (T)
{
exp L [
N (
tp T)] L [
N T I
=
+
(
)
820] - L [
N (
tp
)
820]}
éq 3.2.2-1
where: T I (T) and tp (T) indicate the reciprocal functions of T I T
() and Tp T ().
In fact, one has more easily information relating to speeds of cooling of
metallurgical thermo stories of diagrams TRC that at moments of crossing of
the isotherm 820 °C. It is in particular the case of steels of welding, whose diagrams TRC are
layouts in a reference mark “speed of cooling with 700 °C-temperature”. Taking into account [éq 3.2.2-1],
one can then express the moment of crossing of the isotherm 820 °C according to Tp T
() and of
!Ti (700) and one obtain like Ti T characterization ():
T I (T)
[
exp F (T)
F (
)
700
L (
N T I
=
-
-
! (
)
700 F (
)
700)]
éq 3.2.2-2
1
With F (T) = 1 [
N T (
p T)] and, in particular!Ti (T (T)) = F (T) Ti (T)
Concretely, one interpolates the function F (T) by a polynomial of degree 5. A thermal evolution
experimental thus is completely defined by the data of the coefficients of the characterizing polynomial
its pilot curve and by its speed of cooling with 700 °C. Validation of this method of
parameterization of the thermal stories “read” on diagrams TRC is presented in [bib2]. In
the unit, and taking into account the relative inaccuracies of the layout of diagrams TRC, the reading of
T I T
() and of the determination of!Ti ()
700, the agreement between the thermal stories read and recomputed
seem very sufficient.
If one has the recordings of the thermal evolutions of diagrams TRC, one can define
each experimental thermal evolution by considering that it is its clean curved pilot. By
elsewhere, if dilatometric tests defining diagram TRC used for
the identification of the model are carried out with constant speeds of cooling, one characterizes
these kinetics of cooling only by their speeds of cooling to 700 °C and one
identically null function F.
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Initial temperatures and end of transformation
A diagram TRC provides, for a series of known thermal stories, the proportions of
various metallurgical components which were formed during cooling as well as
temperatures for which one observes on a swelling behavior a significant variation of
total coefficient of dilation of the test-tube [Figure 2.1-a]. These temperatures are then considered
like and the end initial temperatures of the transformations. More precisely:
· initial temperatures of transformation indicated on diagrams TRC
correspond to 1% of component already formed;
· the temperatures of end of transformation correspond to the final proportion of the component
in the course of formation minus 1%.
Kinetics of the ferritic, perlitic and bainitic transformations
The observation of a swelling behavior shows that, except in the vicinity of initial temperatures and
of end of transformation, the evolution of the deformation according to the temperature is almost
linear. Taking into account the equation [éq 2.1-1] evolution of the quantity of phase transformed into
function of the temperature is then not very distant from a function refines and one thus supposes only:
· for the ferritic, perlitic and bainitic transformations, the speed of transformation is,
between the experimental temperatures of beginning and end of transformation, a function
linear of the temperature;
· speeds of these transformations are twice slower at the beginning (from 0 to 1% of
component transformed) and into end of transformation (of Z
- 1% with Z
final
final) that enters them
experimental temperatures of beginning and end of transformation.
Martensitic transformations
It is supposed that the martensitic transformations are described by the law of Koistinen-Marburger
[éq 2.3-3] and the phenomenologic equation [éq 2.3-4] expressing ms. One uses then each
diagram TRC to determine the coefficients, A and Zs as well as the Ms0 temperature. Lastly,
to prevent that the model systematically transforms into martensite remaining austenite when one
reached temperature ms, one introduces an additional parameter, called TPLM, characterizing (by
its slowest speed of cooling with 700 °C) of the kinetics of cooling which generates
a martensitic transformation. More precisely [Figure 3.2.2-a]:
·
Ms0 is regarded as the martensitic initial temperature of transformation when
this one is total;
· is supposed to be constant and calculated in order to check, in the case of a transformation
martensitic total:
Z (MF
4
) = 0 9,9
where MF is the experimental temperature of end of transformation;
· finally, A and Zs are determined by linear regression starting from the stories
thermo metallurgical experimental leading to a martensitic transformation
partial.
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Mso = 415°C; Mso - MF = 185°C
Mso - ms according to (Zf+Zp+Zb)
from where a value of (supposed
Mso - ms
constant for a size of grain
20
Linear (Mso - ms)
data) of - 0.0249
y = 30,086x - 11,437
15
R2 = 0,8648
With = - 30,086
10
Zs = 0,38
Mso - ms (°C)
5
and TPLM = - 9°C/S
0
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
Zf+Zp+Zb
Appear 3.2.2-a: Evolution of (Ms0 - ms) according to (Z1 + Z2 + Z3);
for steel 16MND5 austenitized 5 minutes with 900 °C.
3.2.3 Effect of the austenitic size of grain on the kinetics of the transformations with
cooling.
The transformations of phase proceed by germination and growth. The stage of germination is done
primarily on the grain boundaries. The size of grain of austenite thus plays an important part on
transformations with cooling. For this reason diagrams TRC are established for
conditions of austenitization given and should not in any rigor be used only for
similar conditions of austenitization. The experimental results tend to show that the size of
austenitic grain modifies more the kinetics of transformation than the end and initial temperature
transformations, which results relatively well in a translation of diagram TRC according to
the axis of times. With each point M of a diagram TRC corresponds the tuple (T, T!,)
Z. To relocate it
TRC according to the axis of times amounts multiplying!
T by a coefficient different from the unit (the axis of times
is given in logarithmic scale) [bib15]. One thus defines a speed of cooling “effective”
!Teff:
!Teff =!T exp (has (D - D))
ref.
with
dref: cut austenitic of reference of diagram TRC, homogeneous grain with a longor.
a: homogeneous coefficient material contrary a length.
The law of evolution retained is thus written:
+
.
T - Ms
!(
Z T) = F (T!T, Z ms)
[
]
= F T, T, Z
with Z = {Z, Z, Z
EFF
EFF
1
2
3}
.
T - Ms
This writing has the advantage of limiting the interpolation to only one diagram TRC, of reference.
Handbook of Référence
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Titrate:
Models of metallurgical behavior of steels
Date:
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Author (S):
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Key:
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Note:
Assumption: the germination and the growth of martensite are regarded as
instantaneous and the density of the sites of nucleation has little influence on this transformation.
The effect of the size of grain thus does not relate to the evolution of the martensitic phase and carries
here only on the evolution of the phases ferritic, perlitic and bainitic.
3.2.4 Data entry of diagrams TRC
Taking into account the preceding assumptions, experimental acquisition data contained in
diagrams TRC thus includes/understands:
· for diagram TRC:
-
the value of the austenitic size of grain dref of the diagram and which will be the size of grain
of reference,
-
the coefficient of translation has for the taking into account of the effect of the size of grain
austenitic,
-
the initial temperature of total martensitic transformation Ms0,
-
the value of the coefficient of the law of Koistinen-Marburger,
-
the value of coefficients A and Z S intervening in the equation [éq 2.3-4],
- values of the six coefficients of the polynomial of degree five interpolating the function
[
ln (
tp T)] (if the thermal stories explicitly are known, each one of them is
regarded as being its clean curved pilot and the definition of his six coefficients is
to renew for each history);
· for each thermal history of a diagram TRC:
-
the speed of cooling with 700 °C,
-
final proportions of ferrite, pearlite and bainite (Zff, Zfp, Zfb),
Tdf
(, Tdp, Tdb)
-
initial temperatures of each transformation
,
Tff
(, Tfp, Tfb)
-
and temperatures of end of each transformation
.
The data entry of a diagram TRC is realizable by a software of data entry (available on workstation).
It is a simple and fast operation (approximately an hour for the data entry of about fifty stories).
The result of this procedure of data entry of diagrams TRC (cf [§An1]) is directly insertable
in a command file of Code_Aster as a command DEFI_TRC, command which
thus contains the data identifying the metallurgical behavior of steel.
The complete definition of the models of metallurgical behavior (values of the Ar3 parameters
“quasi static” temperature of ferritic transformation, Ms0, and the complete definition of the model
with the heating and of austenitic growth of grain) is realized within the command
DEFI_MATERIAU under the key word factor META_ACIER.
An example of the procedure of data entry is presented in appendix [§An1].
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3.3 Evaluation of the function of evolution starting from the data
experimental
3.3.1 Evaluation of the function of evolution for the experimental stories
Taking into account the assumptions concerning the evolution of the structure transformations associated
metallurgical stories thermo Hi of a diagram TRC, one thus has a whole of solutions
particular parameterized by dref of the differential equation (for T ms):
!(
Z T) = F (T!T, Z; dref)
who allow for any state thermo metallurgical Ek = {T, T!, Z; dref} of an experimental history
Hi to calculate:
!
Z (E (T)) = F (E
K
K)
Indeed:
Z
D
!
Z (E (T)) =
(E)!T (E
K
K
K)
dT
however, taking into account the assumptions of linearity on the evolutions of Zi T
() between two consecutive states
I
I + 1
E and
of the same discretized history:
K
Ek
dz (
I
I
Z
+1
E K - Z E K
E K)
() (
)
=
dT
T (I
I + 1
E K) - T (E K
)
where T (Ek) can be estimated by derivation of the analytical expression selected to represent Ti T ().
Thus, one can, for any temperature T, to know the values taken by the function F in the states
thermo metallurgical I.E.(internal excitation) = {T, iT! (T), zi (T); dref} where index I refers to the stories
known in experiments.
3.3.2 Calculation of the advance of the transformations for an unspecified state
It acts, knowing T, T!,
Z ms and D at one moment T given, to determine the values of the variables
metallurgical at the moment (T + T) according to. More precisely:
· If
T (T) Ar3 or if!T > 0,
the metallurgical model of transformation to cooling is inactive [§4.4].
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· If
Ar
T (T)
(
Ms T
3 >
),
!(
Z T) = F (T!T, Z; D) = F (T!T, Z; D
EFF
Z T + T
= Z T + Z! T T
ref.) and (
)
() ()
then:
I =
+
3
(
Ms T
T)
Ms
With
Z
0
(T + T) - Zs
+
=
+
I
I
= 1
and, if T (T + T) M (
S T + T): Z (T + T) = Z
4
4 (T)
or, if not:
I = 3
+
Z
4 (T +
T) = 1 -
Z (T + T) 1
- exp [M (St + T) - T (T + T
I
)]
I = 1
· If
T (T) < M (
S T)
(zt + T) = (zt); M (St + T) = M (St)
and
I = 3
+
Z (T + T) = 1
- Z (T + T) 1
-
exp (cd.
4
) [M (St + T) - T (T + T
I
)].
I = 1
If Ar
T (T)
(
Ms T
3 >
), one determines (thanks to the assumption of regularity of F) the value
catch by F in (T, T!, Z; D) starting from knowledge for any temperature T of the values taken
by F in metallurgical states thermo I.E.(internal excitation) {T, iT! (T), zi (T); dref (T)} known stories
.
in experiments, where T (T
I
) is the speed of cooling for the Hi history at the temperature T
(obtained by interpolation). More precisely, one will determine a linear approximation of F with
vicinity of (T, T!, Z; D). F is related to 5 (because the dependence compared to the parameter D
is included in the possible modification the current speed of cooling [§ 3.2.3]) in,
to determine a linear approximation of F in the vicinity of (T, T!, Z; D) amounts determining
the equation of a hyperplane in 6 and thus to have the value taken by F in six points
{E, F (E) “close” to (T, T!, Z; D).
I
I}
Concretely, stages of this interpolation of the values of F in (T, T!, Z; D) are them
following:
· one calculates an “effective” temperature!
Teff allowing to take account of the effect of the size
of austenitic grain if it is different from that of the diagram, and the value then is sought
catch by F in (T, T! , Z; D
EFF
ref.)
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· one calculates for all the experimental stories Hi known the values taken by
function F in the thermo states metallurgical following (in order to know a whole of
values of F in a vicinity of (T, T! , Z; D
EFF
ref.) rather dense in temperature):
1
E
T
=
T, T! T, Z T; D
T
I ()
{I () I () ref. ()}
2
E
T
=
T + 5 °C, T! T + 5 °C, Z T + 5 °C; D
T + 5 °C;
I ()
{
I (
) I (
) ref. (
)}
3
E
T
=
T - 5 °C, T! T - 5 °C, Z T - 5 °C; D
T - 5 °C;
I ()
{
I (
) I (
) ref. (
)}
· one determines the six closer neighbors of E (T) = {T (T), T! (T), (zt); D (T
EFF
ref.
)} among all
J
I.E.(internal excitation) (T) (J = 1)
3
, defining the metallurgical behavior of material in the vicinity of
J
the temperature T (T) by minimizing the distance from E (T) to each one of I.E.(internal excitation) (T);
· one calculates the barycentric co-ordinates of E (T) compared to his closer neighbors
v
E (T) (v = 1)
6
. For that, one solves the linear system associated with this calculation within the meaning of
least squares and by choosing the solution of minimal standard if sound
determinant is null (it is the case when the closest neighbors belong to a variety
closely connected of size lower than six - [R6.03.01]);
· only the neighbors W are retained
E (T) (W)
6 such as all the barycentric co-ordinates
W of E (T) are positive (so that E (T) is located inside the convex polyhedron
being based on these points);
· one calculates then:
!
Z (
W
E (T)) = F (E) =. F [E
K
K
W
(T)] /W
;
W
· finally, one calculates Z with the step of time following Z
(T (+ T)) according to the diagram clarifies according to:
(zt + T) = (zt) +!(zt) T.
Note:
The definition of a distance used in the criterion of proximity is not obvious, account
held of the nonadimensional character of the space of the {T, T!, Z; D}. Currently,
seek closer neighbors is carried out by adimensionnalisant each one simply of
variables but one could plan to introduce weighting coefficients into each
“direction” (T, T!, or)
Z in order to return account of a dominating part played by such or such
variable.
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4
Metallurgical model of behavior to the heating
4.1 Assumptions
During the heating, the only transformation likely to occur is the transformation into
austenite, which one supposes speed independent the heating rate. In addition, one
also suppose that the whole of the phases ferritic, perlitic, bainitic and martensitic
transform in an identical way into austenite. These assumptions are generally common to
the whole of the models of austenitization [bib9], [bib10] and [bib11]. Consequently the model selected is
form:
!
Z
= F (T, Z
).
It is pointed out that the metallurgical model of transformation proposed by Leblond and Devaux and established
in the code Sysweld [bib11] is form (for the transformations with the heating and with
cooling):
Z
T - Z
!
eq
Z (T, Z)
()
=
(T)
where, for the austenitic transformation, the parameter is taken constant.
Comparative data to the experiment presented in [bib11], [bib12] and [bib13] show that,
with the help of the identification of functions Z (T
eq
) and (T) starting from tests at various speeds of
heating, this model allows a completely satisfactory description of the austenitic transformation
steels. Nevertheless, it seems that the identification of the function (T) remains difficult [bib4].
In Code_Aster, the austenitic model of transformation is form:
Z
T - Z
eq
!
Z (T, Z)
()
=
(T)
but with a simple form for the function T
(), in order to keep a whole of models
metallurgical of easy and fast identification.
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4.2
Form model selected
In its continuous form, the model selected is such as:
Z
T - Z
eq
!
Z (T, Z)
()
=
(
éq 4.2-1
T)
where:
·
Z indicates the proportion of austenite;
·
Z (T
eq
) is the function (with Ac1et Ac3 positive constants):
0
if T ac
1
T - Ac
Z
=
1
if
éq 4.2-2
1
eq (T)
Ac
T
Ac
Ac
3
3 - Ac
1
1
if T
Ac3
·
T
() is the function (avec1et 3 positive constants):
if T
1
Ac
1
T - Ac
(T) =
1
1 +
(3 - 1) if Ac1 T ac
Ac
3
éq 4.2-3
3 - Ac
1
if T
3
Ac3
Notice 1: definition of the function Zeq T
()
The definition of the function Zeq T
() is identical to that given by Leblond and Devaux in
[bib11] and [bib12]. It corresponds to the evolution of the austenite rate transformed for
very low heating rates. Indeed, in T fixed, Zeq T
() is the asymptotic solution
towards which the solution of the differential equation [éq 4.2-1] tends with the time-constant
T
(). For low heating rates, the asymptotic solution can be considered
like attack at every moment and Zeq T
() thus corresponds to the evolution of the austenite rate
transform during “quasi-static” evolutions. The function Zeq T
() is thus entirely
defined by the data of Ac1 and Ac3 which is done under single-ended spanner words AC1 and AC3 under
the key word factor META_ACIER of command DEFI_MATERIAU.
Notice 2: form function T
()
In the model suggested by Leblond and Devaux, the form of the function T
() is not
specified and this function is identified in order to obtain a satisfactory agreement between
initial temperatures and end of transformation experimental and calculated. In order to obtain
a model of identification simple and rapid we chose a simple form for the function
T
(). More precisely, to be able to integrate the equation of evolution [éq 4.2-1] there is all
initially considered the case where the function T
() is constant. In this case, one can then
to propose two possibilities of simple identification of this constant function. The first
possibility consists in identifying a value 1 of making it possible to describe it correctly
beginning of the transformations whereas the second consists in identifying a value 3 of
allowing to describe the end of the transformations correctly. The model was then tested
obtained with a function T
() refines definite from values 1 and 3 above definite.
Results obtained being completely satisfactory and comparable with those obtained with
model available in Sysweld, one chose to introduce into Code_Aster a model where
function T
() is closely connected and is defined by 1 and 3 which is indicated with AC1 and AC3.
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4.3
Integration of the equation of evolution
In Code_Aster, one chose to integrate the equation of evolution [éq 4.2-1] exactly in Z and
.
explicitly in T and on each step of time (i.e while considering!
T and constants on the step and
equal to their values at the beginning of step of time). One obtains then:
- T T! T
T T! T
T
Z (T + T)
() ()
=
+ Zeq (T + T)
() ()
- -
+ Zeq (T) - Z
(T) exp-
.
Ac3 - ac
Ac
1
3 - Ac
1
(T)
The consequent evolution of the proportions of all the other metallurgical components is then defined
by:
Z T
+ T
- Z T
Z
I (T + T
)
(
)
()
= Zi (T). 1 -
.
1 - Z (T)
In other words, each phase present is transformed into austenite to the amount of its
proportion at the beginning of step of time.
4.4
Evolution of the austenitic size of grain to the heating
Once austenized, steel sees its size grain to increase more or less quickly according to
temperature, but this growth always takes place since austenite appears with a size of grain
null. The austenitic growth of grain is a thermically activated process. The model of
growth selected is that of Grey and Higgins, adapted to treat material in the course of
transformation [bib15]:
Model of growth:
D
1 1
1
(D) = -
dt
D dlim
Growth in the course of transformation, austenite appearing with a null size of grain:
D
1 1
1
dz/dt
(D) = -
-
D
dt
D D
Z
lim
with
Qapp
= 0
exp (
)
RT
Wapp
dlim = d10 exp (-
)
RT
with
Z: proportion of the austenitic phase
D: diameter of austenitic grain homogeneous to a length
dlim: cut limiting grain, dependant on D homogeneous parameter material to a length
10
Q and W: homogeneous parameters materials with energies of activation (J.mol-1)
app
app
R: constant of perfect gases (8.314 J.K-1.mol-1)
D: homogeneous parameter material at seconds per unit of length
10
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Note:
The parameters materials are to be informed under key word META_ACIER of
DEFI_MATERIAU.
4.4.1 Processing
numerical
The calculation of the size of grain is carried out after the calculation of proportion of phase and the integration of
the equation of evolution is made according to an implicit scheme in D.D' où:
1 1
1
Z
D = -
T -
D
D dlim
Z +
1 1
1
Z
D = D - + D
= D - + -
T
-
D
D Dlim
Z +
A quadratic equation in D. is solved.
4.5
Feel metallurgical evolution
In a structural analysis, certain zones can undergo a heating while others
cool. Moreover, under certain conditions, an austenitic transformation initiated at the time of
heating can continue at the beginning of cooling. There thus does not exist, strictly speaking, one
austenitic model of transformation and a model of transformation to cooling but only one
model of metallurgical transformations which according to the temperature considered and signs it speed
of thermal evolution is described either by the model of decomposition of austenite, or by the model
of formation of austenite.
With regard to the model introduces into Code_Aster, the direction of the metallurgical evolution
(i.e. formation or decomposition of austenite) is given as follows:
T T
(+ T) < Ac1
[Ac
]
> Ar
1; Ar3
3
.
T T
() > 0
AUST
.
T T
() = 0
REFR
if Z Zeq REFR if Z < Zeq AUST
AUST
.
T T
() < 0
REFR
AUST
where REFR means that the metallurgical evolution is determined by the model of decomposition of
the austenite and where AUST means that the metallurgical evolution is determined by the model of formation
austenite.
Note:
AR3 is also a characteristic of the metallurgical behavior to cooling
already defined by the model of transformation in cooling.
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5
Identification of the model to the heating
5.1
Determination of the function Zeq (T)
Zeq T
() can be regarded as the quasi static solution of the differential equation [éq 4.2-1] and
one chooses to define it (as in [bib11]) by the relation [éq 4.2-2].
In this expression, temperatures AC1 and AC3 are the quasi static temperatures “theoretical”
of beginning and austenitic end of transformation which correspond toan austenite rate still formed
equal to zero or already equal to one.
In fact, these temperatures are difficult to determine in experiments and one generally considers
that quasi static temperatures of beginning and given end of austenitic transformation
in experiments correspond, respectively, to 5 and 95% of formed austenite. In other words, if
one notes Ac'1 and Ac' 3 these temperatures, they check:
Z (Ac') =,
0 05 and Z (ac
eq
eq
') =,
1
3
0 95
éq 5.1-1
To determine Ac'1 and Ac' 3 one can use tests of dilatometry at low heating rate
or to apply formulas of the literature connecting the quasi static temperatures of beginning and end of
austenitic transformation with the composition of steels. In general these temperatures are also
indicated on diagrams TRC used for the identification of the model of transformation to
cooling or can be considered using formulas knowing the composition of steel
[bib4].
Lastly, knowing Ac'1 and Ac' 3, one can then determine the temperatures Ac1 and defining Ac3
function Zeq T
() starting from the two equations [éq 5.1-1] above. A complete example of identification
austenitic model of transformation is presented in [bib4].
5.2
Determination of function TAUX (T)
In a general way, it is not easy to release means of a simple and fast identification of
function T
(). This is why one proposes to adopt for this function the simplified form
below [éq 5.2-1] if ac T ac:
1
3
T - Ac
(T) =
1
1 +
(3 - 1)
éq 5.2-1
Ac3 - Ac1
where 1 and 3 is positive constants.
For the phase of identification, one considers the particular case initially where is constant enters
Ac and ac. One proposes two types of identification then allowing to determine is a value
1
3
1
of coherent with the experimental temperatures of beginning of transformation, that is to say a value 3
of coherent with the experimental temperatures of end of transformation.
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One presents in [bib4] the results obtained by these two identifications and one shows (without another form
of theoretical justification) that the function T
() refines definite with values 1 and 3 previously
determined allows to obtain an agreement with the experiment completely comparable with that obtained with
model of Leblond.
5.2.1 Identification of TAUX_3 starting from AC'3
For!
T and constants and initial condition Z (Ac1) = 0, the solution of the equation
of evolution [éq 4.2-1] is (as long as Z (T
eq
) is constant, i.e. as long as T Ac3):
Ac1 - T
Z (T) = Z
.
. !. 1 exp
eq (T) -
Zeq (T)
T
-
. T!
In particular, one thus has, for T
= A
C 3:
Ac - Ac
0 95 = Zeq (ac) -. Z eq (T). !T. 1
1
3
,
3
- exp
. !
T
A test of dilatometry at heating rate constant (and not very low) allows then
to determine value 3 of allowing to reach the agreement between the experimental values and
calculated of Ac3. One presents in [bib4] comparisons between experiment and calculation obtained in
thus identifying the function considered as constant.
5.2.2 Identification of TAUX_1 starting from AC'1
In the same way that previously, one can also write, for T = ac:
1
Ac - Ac
0 05 = Zeq (ac) -. Zeq (T). !T. 1
1
1
,
1
- exp
éq 5.2.2-1
. !
T
There still, having a test at constant heating rate, the equation [éq 5.2.2-1] allows
to determine a value 1 of allowing to obtain a good agreement on the Ac1 temperatures
calculated and experimental.
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6
Model of calculation of hardness
Metallurgical calculations can be supplemented by a calculation of hardness of “hardening” associated with
metallurgical structure.
The selected model uses the assumption according to which the hardness of a polyphase material point is well
represented by a linear law of mixture of the microhardnesses of the components (here phases austenite,
ferrite, pearlite, bainite and martensite). The microhardnesses are taken as being constants of
material and of the phase considered.
The model is written then: HV = Z HV
K
K
K
HV: hardness (here Vickers for example) of the polyphase point,
zk: proportion of the phase K,
HVk: hardness of the phase K.
Although enough simple, this model gives very correct results [bib14].
In Code_Aster the calculation of hardness is done via the operator of postprocessing
CALC_ELEM; option “DURT_ELGA_META” for calculations of hardness at the points of Gauss and option
“DURT_ELNO_META” for calculations with the nodes by elements.
Hardnesses of the various metallurgical phases are data materials provided by the user
under the key word factor DURT_META of operator DEFI_MATERIAU.
DURT_META
(
F1_DURT: HVf
F2_DURT: HVp
F3_DURT: HVf
F4_DURT: HVf
C_DURT: HV
)
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Version
5.0
Titrate:
Models of metallurgical behavior of steels
Date:
29/03/01
Author (S):
F. WAECKEL, A. RAZAKANAIVO
Key:
R4.04.01-C Page:
24/28
7 Bibliography
[1]
WAECKEL F.: Metallurgy elements for the study of structure transformations in
steels. Note Intern EDF-DER, HI-71/8075, March 1993.
[2]
WAECKEL F.: A law of thermo behavior metallurgical of steels for calculation
mechanics of the structures. Thesis of Doctorat be Mécanique, Ensam Paris, March 1994.
[3]
ANDRIEUX S., WAECKEL F.: A metallurgical model of behavior of steels with
cooling. Note intern EDF-DER, HI-74/94/072/0, October 1994.
[4]
WAECKEL F.: Modeling of the austenitic transformation in Code_Aster. Note
intern EDF-DER HI-74/95/017/0, July 1995.
[5]
WAECKEL F., DECEIVED P.: Comparison of the metallurgical models of behavior
established in Code_Aster and sysweld. Note intern EDF-DER, HI-74/94/076/0 and
HT-26/94/033/A, March 1995.
[6]
MARIGO
J.J. :
The thermomechanical behavior of the simple mediums with variables
mémoratrices: I Generalities and first examples. Note Intern EDF-DER HI/5147-07, 1985.
[7]
KOISTINEN DP, MARBURGER RE: With general equation prescribing extent austenite off -
martensite transformation in pure Fe-C Al and lime pit carbon steels. Acta Metallurgica,
vol.7, pp. 59-60, 1959.
[8]
AEBY-GAUTIER E.: Transformations perlitic and martensitic under tensile stress
in the aciers.Thèse of science Doctorat Physiques, Institut National Polytechnique of
Lorraine, 1985.
[9]
DENIS S. FARIAIS D. SIMON: With, Mathematical model coupling phase transformations and
temperature evolutions in steels. ISIJ International, vol.32, n° 3, 1992, pp. 316-325.
[10]
GIUSTI J.: Constraints and residual deformations of thermal origin. Application to
welding and with the hardening of steels. Thesis of Doctorat of State be Science Physique. University
Paris 6, 1981.
[11]
LEBLOND
JB,
DEVAUX
JC: With new kinetic model for anisothermal metallurgical
transformation in steels including effect off austenitic grain size, 1984. Acta Metallurgica,
vol.32, n° 1, pp. 137-146.
[12]
J.C. DEVAUX: Studies of the constraints of thermal origin in the zones affected by
heat at the time of an operation of soudage.Contrat D.G.R.S.T., n° 79.1095, Account-returned of end
of search.
[13]
LEBLOND J.B., MOTTET G., DEVAUX J., DEVAUX J.C: 1985. Mathematical models off
anisothermal phase transformation in steels and predicted plastic behavior. Materials
Science and Technology, vol.1, n° 10 october 1985, pp. 815-822.
[14]
RAZAKANAIVO A.: Introduction of a model of calculation of hardness into Code_Aster.
Report MN, 97/005, January 1997.
[15]
MARTINEZ Mr.: Junction 16MND5-Inconel 690-316LN by welding-diffusion, development and
calculation of the residual stresses of process. Thesis of Doctorat Mines of Paris, Décembre
1999.
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
Code_Aster ®
Version
5.0
Titrate:
Models of metallurgical behavior of steels
Date:
29/03/01
Author (S):
F. WAECKEL, A. RAZAKANAIVO
Key:
R4.04.01-C Page:
25/28
Appendix 1
One presents below an example of the procedure of data entry of a metallurgical thermo history
experimental (at nonconstant speed of cooling). Provided information is first of all shown
by the user with the software of data entry: speed of cooling with 700 °C, composition with the ambient one,
experimental temperatures of beginning and end of transformation,…. This information is registered in
boldface and Italic characters; the temperatures are indicated in °C and speeds of cooling in °C/S.
The whole of the metallurgical thermo states defining this experimental history and their storage
data processing are then presented and the figure [Annexe1-a] represents, always for this history and in
function of the temperature, supposed metallurgical evolution solution of the differential equation [éq 2.3-2] thus
that the recall of the data provided during the data entry.
DATA ENTRY OF THE TRC
DATA ENTRY OF THE STORIES THERMOMETALLURGIQUES OF WHICH ONE A
THE EXPERIMENTAL KNOWLEDGE FOR A STEEL GIVES
Enter the name of steel (8 alphabetical characters maximum)
trcacier
Enter the value of the Ar3 temperature
836
To enter the value of the temperature in lower part of which all them
transformations are finished
200
THE TRC EAST CHARACTERIZES BY:
1 - The number of stories which composes it;
2 - The coefficients A, B, C, D, E, F of the polynomial: WITH + BT + CT2 + DT3 + ET4 + FT5
defining the pilot curve of cooling F (T) such as:
T (T) = exp {F (T) - F (700) - ln [Tp (700) F' (700) 1}
where: Tp is the derivative of T (T) and F' that of F;
3 - The value of the austenitic size of grain dref of the diagram;
ATTENTION! THE FIRST SEIZED HISTORY MUST BE SLOWEST;
I.E. NEAREST TO THERMODYNAMIC BALANCE
Enter the number of sets which you want to seize?
1
ATTENTION! you will seize 1
together (S) of metallurgical thermo stories
(OK = 0; Not = 1)
0
Enter the number of stories of unit 1
1
Enter the value of coefficients A, B, C, D, E, F and D
8.563 - 0.0276 1.22D-4 - 2.955D-7 3.492D-10 - 1.517D-13 11D-6
ATTENTION! the significant minimal proportion for a component with the ambient one
is 0.03
History number 1
Enter the value of Tpoint 700
- 0.00542
Enter the final proportions of ferrite, pearlite and bainite (Zff, Zfp and Zfb)
for history 1
0.764 0.199 0.037
Input of and the end initial temperatures of transformation for history 1
Enter and the ferritic end initial temperatures of transformation Tdf and Tff
792 657.5
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
Code_Aster ®
Version
5.0
Titrate:
Models of metallurgical behavior of steels
Date:
29/03/01
Author (S):
F. WAECKEL, A. RAZAKANAIVO
Key:
R4.04.01-C Page:
26/28
Enter and the perlitic end initial temperatures of transformation Tdp and Tfp
657.5 615
Enter and the bainitic end initial temperatures of transformation Tdb and Tfb
490 420
VALIDATE you the HISTORY NUMBER? (YES = 1 NOT = 0) 1
TPOINT A 700 DEGREES
- 5.420D-03
Zff
Tdf
Tff
7.640D-1
7.920D+2
6.575D+2
Zfp
Tdp
Tfp
1.990D-01
6.575D+2
6.150D+2
Zfb
Tdb
Tfb
3.700D-02
4.900D+2
4.200D+2
1
DEFINITION of the coefficient of translation used to calculate!
Teff, modelling
the influence of the size of grain on the kinetics of transforamtion:
!Teff =!T exp (has (D - D))
ref.
The value of A. is thus defined.
Enter the value of the coefficient of translation has for the effect cuts grain:
11200.
DEFINITION OF the VARIATION OF ms According to Zf + Zp + Zb
It is considered that the martensitic transformation is described by the law of
Koistinen-Marburger:
I = 3
Z
+
4 (T)
=
1 -
Z
I - exp
-
{1
([Ms T])}
I = 1
in which the martensitic initial temperature of transformation ms is, with
beyond certain threshold, function of Zf + Zp + Zb:
+
I
= 3
(
Ms T)
Ms
With
Z
0
- Zs
=
+
I
.
I
= 1
One thus defines the values of Ms0 and, as well as the values of Zs and A.
Enter the number of laws of variation of ms according to Zf + Zp + Zb that you
please seize
1
Enter the value of the Zs threshold and A for law 1
as well as the value, TPLM, speed of cooling with 700 °C of
the slowest history leading to a martensitic transformation
partial and of
0.47 - 32.76 - 3,497 14.06
You validate the law such as (OUI = 1 NON = 0): Zs = 0.47
AM = - 32.76
TPLM = - 3,497
= 14.06
1
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
Code_Aster ®
Version
5.0
Titrate:
Models of metallurgical behavior of steels
Date:
29/03/01
Author (S):
F. WAECKEL, A. RAZAKANAIVO
Key:
R4.04.01-C Page:
27/28
Example of procedure of data entry of diagrams TRC
tracier = DEFI_TRC
(- 5.420D-03
1.100D+01
8.563D+00
- 2.760D-02
(HIST_EXP:
VALE:
1.220D-04
- 2.955D-07
3.492D-10
- 1.517D-13
0.000D+00
0.000D+00
0.000D+00
8.360D+02
0.000D+00
0.000D+00
0.000D+00
7.956D+02
1.000D-02
0.000D+00
0.000D+00
7.920D+02
7.277D-01
0.000D+00
0.000D+00
6.622D+02
7.540D-01
1.000D-02
0.000D+00
6.575D+02
7.640D-01
2.523D-02
0.000D+00
6.539D+02
7.640D-01
1.890D-01
0.000D+00
6.150D+02
7.640D-01
1.990D-01
0.000D+00
6.103D+02
7.640D-01
1.990D-01
0.000D+00
5.665D+02
7.640D-01
1.990D-01
1.000D-02
4.900D+02
7.640D-01
1.990D-01
2.700D-02
4.250D+02
7.640D-01
1.990D-01
3.700D-02
3.485D+02)
TEMP_MS: (P: 1.100D+01
SEUIL: 4.700D-01
AKM: - 3.276D+01
BKM .....:1.406D+01
TPLM: - 3.497D+00)
GRAIN_AUST: (DREF: 11.D-6
A: 11200.)
;
Result of the operation of data entry above providing in language
order Aster the definition of a metallurgical behavior with
cooling.
Visualization of the points seized and calculated for a history
of a TRC
80
70
Zf (T) built
60
Zp (T) built
50
Zb (T) built
40
30
pts Zf-T seized
20
pts Zp-T seized
10
pts Zb-T seized
0
200
300
400
500
600
700
800
900
Temperature (°C)
Appear Annexe1-a: Exemple of thermo history metallurgical resulting from a TRC
and integrated into the model
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
Code_Aster ®
Version
5.0
Titrate:
Models of metallurgical behavior of steels
Date:
29/03/01
Author (S):
F. WAECKEL, A. RAZAKANAIVO
Key:
R4.04.01-C Page:
28/28
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Handbook of Référence
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
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