APPROXIMATE MODEL FOR PREDICTING STATIC
RECRYSTALLIZATION OF FERRITIC STAINLESS STEEL TYPE 430
R. Benchouieb1, D. Berdjane1, S. Achouri1, O.Ghelloudj1, F. Lemboub1
1Research Center in Industrial Technologies CRTI, P.O.BOX 64, Cheraga 16014 Algiers, Algeria
ABSTRACT
An approximate model for predicting static recrystallisation of ferritic stainless steel type 430, in hot rolling is proposed. In this model, the effect of variables such as strain, strain rate, temperature and initial grain size were considered during hot rolling operations.
A set of integrated mathematical models for predicting static recrystallisation evolution during hot rolling has been developed through laboratory research work experiments. It consists of many sub-models such as percentage of recovery, recrystallisation kinetics, time for 50 percent of recrystallisation, recrystallised grain size and grain growth. Some of the most important theoretic basic approaches to describe the kinetics of primary recrystallisation were first independently developed and comprehensive portrayed by Johnson and Mehl, Avrami as by Kolmogorov (Often named the JMAK-theory).
The quantitative determination of the effects of these variables obtained, analysed and compared in the context of the recrystallisation kinetics of this material.
The predicted results are in good agreement with measured of laboratory tests. The results of constitutive models based on semi empirical equations will be compared in the next publication to more sophisticated models based on cellular automata, vertex and Monte- Carlo-Potts methods.
Keywords: Ferritic stainless steel, Hot rolling, Static recrystallisation, Mathematical model.
1. INTRODUCTION
The microstructure evolution during hot rolling is of great interest for the industrial production of steel and has therefore been subject of research in the past decades.
Recrystallisation during and after rolling is one of the mechanisms that can be used for grain refinement [1].
In consequence, controlling the recrystallisation is one opportunity for controlling the mechanical properties and in turn gives room for saving expensive alloying elements. Therefore models with good predictive capabilities are necessary.
Since dynamic recrystallisation had been observed by Greenwood and Worner [2], many theories and models for the description of recrystallisation during and after deformation have been developed. These models differ in terms of complexity, characteristic length scale, practical usability and the considered materials.Additionally the usability of the different model types for technical applications, like process control or numerical simulations of hot forming processes, was discussed. One important aspect is the required experimental effort necessary for determining material parameters.
Although there is a great deal of empirical knowledge of the microstructures that can be produced during current industrial processing, the ability to produce more nearly ideal microstructure for different applications is very limited and it is in order to gain improved control of recrystallization processing that increased scientific understanding is needed.
It described the sub-models for static softening kinetics, static recovery, static recrystallisation and grain growth. These sub-models can be employed singly or linked in sequence with the output data of one model as input data for the next model.
The studies on the recrystallisation have showed that ferritic stainless steels have a slow recrystallisation kinetics [3,4] and consequently they present a good choice to
make experiments without the intervention of the recrystallisation.
This study was made on a ferritic stainless steel type 430 material by using the hot rolling experiments and the quantitative metallography techniques with one pass only.The effect of variables such as strain, strain rate, temperature and initial grain size were quantified for studying microstructural evolution during hot rolling operations.
The comparisons of calculated with measured parameters such as volume fraction recrystallized, recrystallized grain and grain growth will be presented.
2. MATERIALS AND METHODS This ferriticstainless steel type 430 was prepared and air-melted, it was cast in two hot topped ingots of 76mm in diameter.
The chemical composition of the ingot obtained is shown in table (1).
Element Weight(%) massique
C 0,001
S 0,008
P <0,01
Mo <0,02
Ni <0,02
Si 0,09
V <0,02
Cr 16,7
Mn 0,18
Nb <0,02
Ti <0,02
Co <0,02
Cu <0,02
Table (1). Chemical composition of the ferritic stainless steel type 430 employed.
2.1. Hot rolling experiments
Hot rolling experiments were performed with the laboratory 2-high, 50 tonnes rolling mill. The slab size was usually 120
×30×15mm. The grain sizes before rolling were examined and found to be 790µm and 150µm respectively. Hot rolling was carried out at different temperatures of 8700C, 9500C and 10000C, with different reductionswithin a range of (25%-50%), using different strain rates. The rolling schedules were given in a single pass and then the slabs were water quenched
immediately after the pass deformation to keep the as-deformed structure. After quenching, the slabs were cut into half their usual width and sectioned to provide specimens to determine a full recrystallisation curve from one deformed slab.
After sectioning and linishing, specimens were ground down through a series of successively finer silicon carbide papers, The chemical etch of Kalling’s reagent containing: 20ml HCl, 15ml H2O, 65ml CH3OH, 1g CuCl2, was selected.
3. RESULTS
The activation energy for static recrystallisation was obtained from the Arrhenius plot of t50% versus the reciprocal of annealing temperature derivedand found to be 232kcal/mol.
3.1. Effect of strain and temperature In order to determine the isothermal recrystallization kinetics after deformation at different effective strains and strain rate of 3.3s-1 where dynamic recovery is the only restoration process during deformation, slabs of this material were deformed at different rolling temperatures.
The recrystallisation curves for coarse and fine initial grain sized material show the characteristic sigmoidal form. Effective strains of (0.29-0.73) were chosen to be within the work hardening region.
3.2. Effect of initial grain size
The results of the effect of initial grain size on the isothermal recrystallization kinetics for a rollingtemperature of 870oCwerereproducedfrom the figures of the effect of strain and temperature on the recrystallization kinetics.
3.3. Effect of strain rate
The results from isothermal constant strain rate tests, at an effective strain of 0.58 performed using slabs of initial grain size of 790µm, were presented to determine the relationship between t50%and Zener- Hollomon parameter, Z.
The various relationships for the effect of deformation variables and initial grain size can be combined to contain overall
descriptions of recrystallisation rate and recrystallised grain size in terms of these variables. Thus, the general form of the two equations found in the present work for this type of material are:
(
RT)
z d A
t 0.21exp 232000
2 . 2 0 7 . 1 0 50
−
=
ε−(1)
11 . 4 0 . 0 0 6 . 0 0
−
= B − d z
drex
ε
(2)To check the accuracy of the above equations, all data from the hot rolling experiments were plotted according to the above relationships on Log-Log scales.
The values of the constants A0 and B0in the above equations were defined and are:
21 . 0 2 . 2 10 12 4 . 0 1
−
× −
= S
A µ
11 . 0 4 . 295 0
0 = S−
B µ
It was found that equations (1) and (2) for both t50%, anddrex, give satisfactory representation of the effect of deformation variables and initial grain size on t50%, and drex. The scatter in the results might be due to the measurement of some parameter such as volume fraction recrystallised.
In this dynamically recovered material the effect of effective strain and temperature on the relation between migrating boundary area per unit volume and volume fraction recrystallised has been determined.It was found that the growth rate decreases with annealing time, t, under all the deformation conditions investigated according to a relation of the form;
67 .
−2
∝ t
G (3)
4. STATIC RECRYSTALLIZATION MODEL
4.1. Introduction
Static recrystallisation after hot deformation is very similar to the classical one after cold deformation. After static recovery has proceeded to some degree, a subsequent further and larger decrease in strength occurs, which is indicative of static recrystallisation, it is attributed to the formation and growth of new grains in the statically recovered microstructure.
The first rate law of transformation associated with static recrystallisation, was
derived by Johnson and Mehl [5]. The equation was expected to apply to any phase transformation where there is random nucleation. For the case where the nucleation rate decreases exponentially;
Avrami[6] modified the nucleation approach of Johnson and Mehl to arrive at this developed equation;
(
BtK)
X =1−exp − (4) Where;
X is the volume fraction recrystallised.
t is the period of time in which the sample is held at an annealing temperature.
B and K are empirical constants at a constant temperature and prior strain.
The value of B is dependent upon both a recrystallisation nucleation rate and a grain growth rate, and the time exponent value of K indicates the dimension of grain growth.
The Johnson-Mehl-Avrami-Kolmogorov (JMAK) model is used in many transformation reactions,andmay be described phenomenologically in terms of the constituent nucleation and growthprocesses. A more generaldiscussion of the theory of the
kinetics of transformations may be found in the phasetransformation literature, e.g.
Christian (2002).
It was assumed that nuclei are formed at a rateNand that grains grow into the deformedmaterial at a linear rate G. If the grains are spherical, their volume varies as the cube oftheir diameter, and the fraction of recrystallized material (XV) rises rapidly with time.However, the new grains will eventually impinge on each other and the rate ofrecrystallization will then decrease, tending to zero as XVapproaches 1.The number of nuclei (dN) actually appearing in a time interval (dt) is less than Ndtbecause nuclei cannot form in those parts of the specimen which have alreadyrecrystallized. The number of nuclei which would have appeared in the recrystallizedvolume is NXVdt and therefore the total number of nuclei (dN1)
which would haveformed, including the
‘phantom’ nuclei is given by:
𝑑𝑑𝑁𝑁1 =𝑁𝑁𝑑𝑑𝑁𝑁= 𝑑𝑑𝑁𝑁+𝑁𝑁𝑋𝑋𝑉𝑉𝑑𝑑𝑁𝑁 (5) If the volume of a recrystallizing grain is V at time t, then the fraction of material whichwould have recrystallized if the phantom nuclei were real (XVEX) is known as theextended volume and is given by:
𝑋𝑋𝑉𝑉𝑉𝑉𝑉𝑉 =∫ 𝑉𝑉𝑑𝑑𝑁𝑁t 1
0 (6)
If the incubation time is much less than t, then
𝑉𝑉 =𝑓𝑓𝐺𝐺3𝑁𝑁3(7)
where f is a shape factor (4𝜋𝜋⁄3 forspheres). Thus ;
𝑋𝑋𝑉𝑉𝑉𝑉𝑉𝑉 =𝑓𝑓𝐺𝐺3∫ 𝑁𝑁0𝑡𝑡 3𝑁𝑁𝑑𝑑𝑁𝑁(8) IfN is constant then ;
𝑋𝑋𝑉𝑉𝑉𝑉𝑉𝑉 =𝑓𝑓𝑁𝑁𝐺𝐺3𝑁𝑁4⁄4(9)
During a time interval dt, the extended volume increases by an amount dXVEX. As thefraction of unrecrystallized material is 1-XV, it follows that;
𝑑𝑑𝑋𝑋𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑑𝑑𝑋𝑋𝑉𝑉⁄(1− 𝑋𝑋𝑉𝑉)(10) 𝑋𝑋= � 𝑑𝑑𝑋𝑋𝑉𝑉𝑉𝑉𝑉𝑉
𝑉𝑉𝑉𝑉
0
𝑋𝑋= �(𝑑𝑑𝑋𝑋𝑉𝑉⁄(1− 𝑋𝑋𝑉𝑉))
𝑉𝑉𝑉𝑉
𝑋𝑋= ln(1 (1⁄ 0 − 𝑋𝑋𝑉𝑉)) (11) 𝑋𝑋𝑉𝑉 = 1− 𝑒𝑒𝑒𝑒𝑒𝑒(−𝑋𝑋𝑉𝑉𝑉𝑉𝑉𝑉) (12) Combining equations (9) and (12) ;
𝑋𝑋𝑉𝑉 = 1− 𝑒𝑒𝑒𝑒𝑒𝑒((−𝑓𝑓𝑁𝑁𝐺𝐺3𝑁𝑁4) 4⁄ ) (13) This may be written more generally in the form ;
𝑋𝑋𝑉𝑉 = 1− 𝑒𝑒𝑒𝑒𝑒𝑒(−𝐵𝐵𝑁𝑁𝐾𝐾) (14) Where 𝐵𝐵= −𝑓𝑓𝑁𝑁𝐺𝐺3⁄4, which is often called the Avrami, Johnson–Mehl or JMAK equation.
In the case considered above, in which the growing grains were assumed to grow in threedimensions, the exponent K in equation (14),which we will refer to as the JMAK orAvrami exponent is seen from equation (13) to be 4.
The above treatment assumed thatthe rates of nucleation and growth remained constant during recrystallization. Table (2) considered all the casesfor the explaination
of the nucleation events effectively occur at the start of recrystallization (Site saturation).
Growth dimensionality
Site saturation
Constant nucleation
rate
3-D 3 4
2-D 2 3
1-D 1 2
Table (2). Ideal JMAK exponents.
The above analyses assume that until impingement, the grains grow isotropicallyinthree-dimensions.
It should be noted that the essential feature of the JMAK approach is that the nucleationsites are assumed to be randomly distributed.
Equation (4) can be more usefully expressed in terms of time to obtain a fixed recrystallised fraction.
As mentioned above from the empirical equations found in this research that the recrystallisation rate depends on the stored energy and the density of nucleation sites, and so is a function of the prior deformation variables (
ε
)and (Z) and the original grain size (d0) as well as the temperature of holding.These equations have been reviewed by many researchers.They found that the equations had the same general form;
[
Q RT]
Z d A
t50 = 0ε−m 0p −uexp rex (15) Where;
d0is the original grain size, R is the universal gas constant, T is the annealing temperature,
Qrexis the activation energy forrecrystallisation.
A0, m, p, u are constants.
If Qrex is known as well as the activation energy for deformation, Qdef
thedependence of t50 on Z can readily be found by defining an apparent activation energy, Qapp. Thusfromequation (15):
( )
[
Q uQ RT]
d A
t50= 0ε−m 0pε−uexp rex− def (16) and if Qrex-uQdef=Qapp
then ;
(
Q RT)
d A
t50 = 0ε−m 0pε−uexp app (17)
Since Qapp can be determined from recrystallisation experiments carried out at different temperatures and constant strain rates, so the constant u can be found.A general form of an equation for predicting the recrystallised grain size, drex, was proposed [7,8] ;
U P M
rex B d Z
d = 0ε− 0 − (18)
Where B0,M,Pand U are constants.
In the occurrence of site saturation the recrystallised grain size is predominantly determined by the terms related to nucleation site density such as the strain,
ε
,and the original grain size, d0. This corresponds to a practically known fact that the Zener-Hollomon parameter has a relatively small effect on the recrystallised grain size for ferritic steels.In this paper we only want to give a brief outline of the basic concept behind this type of model. It has to be noted that a multitude of modified or extended versions of the empirical submodels exist in literature. As input parameter these models use temperature, strain and strain rate of each deformation step, the initial grain size and the time after the prior deformation.
The output parameters are the recrystallized volume fraction and the average recrystallized grain size.
Each model contains material parameters to be derived from experiments. The recrystallized volume fraction has to bedetermined after thermomechanical treatments under various processing conditions.
The static recrystallisation behavior of this ferritic stainless steel type 430 can be modelled by the Avramiequation (4).
4.2. Results of the static recrystallization model
The calculation of the volume fraction recrystallized was carried out using the calculated values of the exponent K and the constant B from the Avrami equation (4).
The results of the calculated values of K and Band the measured ones in relation with the deformation parameters and initial
grain size are presented in tables (3) and (4) and illustrated graphically in figures (1-6).
RT
(0C)
d0
(µm) ε 𝜺𝜺 ̇
(s-1)
KC KM
870 790 0.73 3.3 1.98 1.85 870 790 0.58 3.3 1.72 1.55 870 790 0.29 3.3 1.84 1.94 1000 790 0.73 3.3 1.88 1.74 1000 790 0.58 3.3 1.24 1.48 1000 790 0.29 3.3 1.93 2.34 950 790 0.58 5.0 2.64 2.77 950 790 0.58 1.0 2.32 2.43 950 790 0.58 0.1 2.00 2.08 950 150 0.73 3.3 0.85 0.64 950 150 0.58 3.3 0.84 0.69 950 150 0.29 3.3 0.87 0.85 870 150 0.73 3.3 1.11 1.55 870 150 0.58 3.3 1.63 1.43 870 150 0.29 3.3 2.28 1.64
Table (3). Calculated and measured values of the exponent K in relation withdeformation parameters and initial grain size.
RT
(0C) d0
(µm) ε 𝜺𝜺̇
(s-1)
BC BM
870 790 0.73 3.3 7.92×10-7 1.53×10-6 870 790 0.58 3.3 7.93×10-7 3.31×10-6 870 790 0.29 3.3 7.88×10-8 3.56×10-8 100 790 0.73 3.3 6.98×10-5 1.33×10-4 100 790 0.58 3.3 6.15×10-4 1.76×10-4 100 790 0.29 3.3 3.08×10-6 2.29×10-7 950 790 0.58 5.0 1.9×10-7 7.77×10-8 950 790 0.58 1.0 5.2×10-7 2.66×10-7 950 790 0.58 0.1 1.03×10-6 6.04×10-7 950 150 0.73 3.3 0.075 0.15 950 150 0.58 3.3 0.037 0.07 950 150 0.29 3.3 0.011 0.01 870 150 0.73 3.3 0.018 0.002 870 150 0.58 3.3 0.001 0.003 870 150 0.29 3.3 1.19×10-5 2.59×10-4
Table (4). Calculated and measured values of the constant B in relation withdeformation parameters and initial grain size.
Figure (1). The effect of rolling temperature on the calculated and measured values of exponent K, ε̇=3.3s-1, d0=790µm.
Figure (2). The effect of rolling temperature on the calculated and measured values of exponent K, ε̇=3.3s-1, d0=150µm.
Figure (3). The effect of strain rate on the calculated and measured values of exponent K, ε=0.58, RT=9500C, d0=790µm.
Figure (4). The effect of rolling temperature on the calculated and measured values of constant B,
ε̇ =
3.3s-1, d0=790µm.1 1,5 2 2,5
800 900 1000 1100
Calculated and measured K
Rolling temperature (°C) ε=0.73, KC ε=0.73, KM ε=0.58, KC ε=0.58, KM ε=0.29, KC ε=0.29, KM
0,5 1 1,5 2 2,5
800 900 1000
Calculated and measured K
Rolling temperature (°C) ε=0.73, KC ε=0.73, KM ε=0.58, KC ε=0.58, KM ε=0.29, KC ε=0.29, KM
1 2 3
0 1 2 3 4 5 6
Calculated and measured K
Strain rate (s-1)
KC KM
1E-09 1E-08 1E-07 1E-06 1E-05 0,0001 0,001 0,01
800 900 1000 1100
Calculated and measured B
Rolling temperature (°C)
ε=0.73, BC ε=0.73, BM ε=0.58, BC ε=0.58, BM ε=0.29, BC ε=0.29, BM
Figure (5). The effect of rolling temperature on the calculated and measured values of constant B, ε̇=3.3s-1, d0=150µm.
Figure (6). The effect of strain rate on the calculated and measured values of constant B, ε
=
0.58, RT=9500C and d0=790µm.4.3. Effect of deformation variables andinitial grain size onthe calculated and measured volume fraction recrystallized
The effect of effective strain and temperature on the calculated and measured volume fraction recrystallized is shown in figures (7) and (8), d0=790µm.
Figure (7). The effect of effective strainon the calculated and mesured volume fraction recrystallized, RT=8700C, d0=790µm.
Figure (8). The effect of effective strain on the calculated and mesured volume fraction recrystallized, RT=10000C, d0=790µm.
The effect of strain rate on the calculated and mesured volume fraction recrystallized is shown in figure (9).
Figure (9). The effect of strain rate on the calculated and mesured volume fraction recrystallized, RT=9500C, d0=790µm, ε=0.58.
The effect of effective strain and temperature on the calculated and mesured volume fraction recrystallized is shown in figures (10) and (11), d0=150µm.
Figure (10). The effect of effective strain on the calculated and mesured volume fraction recrystallized, RT=9500C, d0=150µm.
1E-06 1E-05 0,0001 0,001 0,01 0,1 1
800 850 900 950 1000
Calculated and measured B
Rolling temperature (°C) ε=0.73, BC ε=0.73, BM ε=0.58, BC ε=0.58, BM ε=0.29, BC ε=0.29, BM
1,00E-09 2,01E-07 4,01E-07 6,01E-07 8,01E-07 1,00E-06 1,20E-06
0 1 2 3 4 5 6
Calculated and measured B
Strain rate (s-1)
BC BM
0 10 20 30 40 50 60 70 80 90 100
0,1 1 10 100 1000 10000 100000
Xv (%)
Effective time (s) ε=0.73, XvC
ε=0.73, XvM ε=0.58, XvC ε=0.58, XvM ε=0.29, XvC ε=0.29, XvM
0 10 20 30 40 50 60 70 80 90 100
1 10 100 1000 10000
Xv (%)
Effective time (s) ε=0.73, XvC
ε=0.73, XvM ε=0.58, XvC ε=0.58, XvM ε=0.29, XvC ε=0.29, XvM
0 10 20 30 40 50 60 70 80 90 100
1 10 100 1000 10000
Xv (%)
Effective time (s) ε̇=5s-1, XvC
ε̇=5s-1, XvM ε̇=1s-1, XvC ε̇=1s-1, XvM ε̇=0.1s-1, XvC ε̇=0.1s-1, XvM
0 10 20 30 40 50 60 70 80 90 100
0,01 0,1 1 10 100 1000
Xv (%)
Effective time (s) ε=0.73, XvC
ε=0.73, XvM ε=0.58, XvC ε=0.58, XvM ε=0.29, XvC ε=0.29, XvM
Figure (11). The effect of effective strain on the calculated and measured volume fraction recrystallized, RT=8700C, d0=150µm.
4.4. Comparison with experiments
Experimental measurements of recrystallization kinetics are usually compared with theJMAK model by plotting ln[ln(1/(1-XV))] against ln(t).
According to equation (14), thisshouldyield a straight line of slope equal to the exponent K. This method of datarepresentation is termed a JMAK plot.
It should incidentally be noted that equation (11) shows that this is equivalent to a plot of ln XVEXagainst ln t.
The growth rate is not easily determined from measurements of XV. However, if the interfacial area between recrystallized and unrecrystallized material per unit volume (SV)is measured as a function of time, then Cahn and Hagel (1960) showed that the globalgrowth rate𝐺𝐺̇ is given by;
𝐺𝐺 =̇ (1⁄𝑆𝑆𝑉𝑉)(𝑑𝑑𝑋𝑋𝑉𝑉⁄𝑑𝑑𝑁𝑁)
It is in fact very unusual to find experimental data which, on detailed analysis, showgood agreement with JMAK kinetics. Either the JMAK plot is non- linear, or the slopeof the JMAK plot is less than 3, or both.
It has been shown in figures (7-11), that for each deformation condition there is no significantchange between the values of the measured and calculated volume fraction recrystallized (XV)as the annealing effective time increases whatever the deformation parameters and initial grain size.
5. DISCUSSION
5.1. Static recrystallisation
In this stainless steel type 430, for all deformation conditions investigated, nucleation is usually associated with original grain boundaries predominantly at grain edges. Barbosa and Santos [9] in material of ferritic stainless steel type 430, after cold rolling by 20 and 40% showed that nucleation takes place at grain boundaries only.
The recrystallisation curves of this fine and coarse grain size material follow the Avrami-Type equation. The time exponent, K in Avramiequation was found to vary between 0.64 and1.64 in material of fine grain size (150µm) and varying between 1.48 and 2.34 in material of coarse grain size (790µm). It was found that K depends on annealing temperature for finer grainmaterial but in coarser grain material the exponent K depends on the effective strain. It was also found that Kdepends on the strain rate varying between 2.08 and 2.77. The values of the exponent K in this ferriticstainless steel type 430 are in good agreement with other reported values for ferritic steels [10,11].
5.1.1.Effect of strain
In this type of material, the negative strain exponent of 1.7 which describes the influence of the amount of prior strain on the rate of recrystallisation is in good agreement with the results of Barbosa and Santos (n=1.7), [9] working with austenitic stainless steel type 304 and ferritic stainless steel type 430 and 430M after cold deformation (430M contains Ti and Nb). The grain size produced after static recrystallisation is dependent on the value of the prior strain, this dependence suggests that the recrystallised grain size is proportional to a strain exponent of -0.6. A strain exponent of -1 has been reported by Barbosa and Sellars [12] working with austenitic stainless steel.
5.1.2. Effect of initial grain size
Initial grain size has been shown to affect both recrystallisation rate, and
0 10 20 30 40 50 60 70 80 90 100
1 10 100 1000
Xv (%)
Effective time (s) ε=0.73, XvC
ε=0.73, XvM ε=0.58, XvC ε=0.58, XvM ε=0.29, XvC ε=0.29, XvM
recrystallised grain size. There is a difference in the rate of recrystallisation with initial grain size for both structures (150µm and 790µm). A value of positive initial grain size exponent of 2.2 was derived, which is a somewhat lower exponent that has been reported previously for steels [13,14]. As for the initial grain size dependence of final grain size, a value of positive initial grain size exponent of 0.4 is in close agreement with those reported previously by Sellars [7]and Castan [15]for single phase material.
5.1.3. Effect of strain rate and tempearature
The effect of raising the strain rate for coarser grain material (790µm),deformed at a temperature of 950oC, is to reduce the time available for competing dynamic restoration during deformation and hence to produce a higher dislocation density within the metal (higher stored energy per unit volume).The accelerating effect on the recrystallisation rate of raising the temperature while holding the strain rate constant has been reported by numerous workers [16] confirming that the amount of stored energy driving the recovery process is progressively reduced and the boundary mobility is increased as the deformation temperature is increased. The recrystallised grain size also increased with deformation temperature, confirming the strong influence of subgrain size and driving force on the processes of nucleation and growth.AZener-Hollomon parameter exponent of -0.21, of the dependence of t50% on Z is lower than those reported previously for ferritic steels[15,17]. A Zener-Hollomon parameter exponent of - 0.11 of the dependence of drex on Z, is in close agreement with those of -0.13, for low alloy steelreported by Sellars [8], however, it is lower than those of -0.54, and -0.36, reported by Kato, Saito and Sakai as well as Fujimura and Tsuge[18,19] working with ferritic stainless steels under different condition of deformation.
5.2.Static recrystallization model
The static recrystallization may be considered as a solid-state transformation and its kinetics can be modeledby the Johnson-Mehl-Avrami-
Kolmogorov(JMAK) equation.
The experimental values of the volume fraction recrystallized were compared to the calculated ones under different deformation conditions using hot rolling experiments. However, the calculation of the volume fraction recrystallized (XvC) was carried out using the calculated values of the exponent K and the constant Bderived from the Avrami equation (9).
From table (3), the results of the calculated and measured values of the exponent Kare presented in table (5) in relation with the hot rolling conditions and their dependence.
Hot rolling conditions
KM, KC Dependence ɛ=0.29
ɛ=0.58 ɛ=0.73 ε̇=3.3s-1 RT=8700C d0=790µm
Very similar K~2
Independent of effective strain
ɛ=0.29 ɛ=0.58 ɛ=0.73 ε̇=3.3s-1 RT=10000C d0=790µm
Nearly equal K~1 K~2
Dependent of effective strain
ɛ=0.29 ɛ=0.58 ɛ=0.73 ε̇=3.3s-1 RT=9500C d0=150µm
Nearly equal K~1
Independent of effective strain
ɛ=0.29 ɛ=0.58 ɛ=0.73 ε̇=3.3s-1 RT=8700C d0=150µm
Nearly equal K~1 K~2
KC dependentof effective strain
ε ̇=0.1s-1 ε̇=1s-1 ε̇=5s-1 ɛ=0.58 RT=9500C d0=790µm
Very similar K~2 K~3
Dependent of strain rate
Table (5). Dependence of KM and KC on the deformation parameters and initial gain size.
The values for the Avrami exponent, K, provide information about the nucleation
mechanisms in the recrystallization event, and following Cahn’s theory, the calculated and measured values of the exponent Kpoint out thatsurface nucleation, edge nucleation and corner nucleationare nucleation mechanisms in the recrystallization event of this ferritic stainless steel type 430, which depend on the deformation parameters.
For low Carbon steels, Avramiexponents, K=1 and K=2, are usually reported[20-24], and it was reported that these values of Kof most materials which happened static recrystallization are equals to 1 and 2, and the deformation variables normally make no difference at that time[25].Some studies however report this Kvalue can be affected by temperature and initial grain size; as the grain size increases or temperature decreases, K value decreases from 2 to 1 [26]. If grain size is especially very fine less than 50µm, the Kvalue is going to be fixed as 1 having no connection with deformation variables or chemical composition [21]. The exponent Kfor recrystallization which typically lies between 1 and 2 in iron and other materials [27-29] has been explained by Doherty[27], this is partly due to the non- random distribution of nucleation sites, which arises from the grain-to-grain variation in the deformation microstructure, which leads in turn to variation in the stored energy on the scale of the grain size. This non-uniformity of the deformation behavior can be related to differences in the dislocation density from grain to grain.
From table (4), the results of the calculated and measured values of the constant B are presented in table (6) in relation with the hot rolling conditions and their dependence.
Hot rolling conditions
BM, BC Dependence
ɛ=0.29 ɛ=0.58 ɛ=0.73 ε̇=3.3s-1 RT=8700C d0=790µm
BM slightly higher than BC
Dependent of effective strain
ɛ=0.29 ɛ=0.58
ɛ=0.73 ε̇=3.3s-1 RT=10000C d0=790µm
BM lower than BC
Dependent of effective strain
ɛ=0.29 ɛ=0.58 ɛ=0.73 ε̇=3.3s-1 RT=9500C d0=150µm
BM higher than BC
Dependent of effective strain
ɛ=0.29 ɛ=0.58 ɛ=0.73 ε̇=3.3s-1 RT=8700C d0=150µm
BM(0.29) higher than BC(0.29) BC(0.73) higher than BM(0.73)
Dependent of effective strain
ε ̇=0.1s-1 ε̇=1s-1 ε̇=5s-1 ɛ=0.58 RT=9500C d0=790µm
BChigher than BM
Dependent of strain rate
Table (6). Dependence of BM and BC on the deformation parameters and initial gain size.
This kind of discrepancy between the measured and calculated exponent K and constant Bdetermined via hot rolling experiments has already been reported in the literature for other steels [30].
6. CONCLUSION
- A model for predicting the recrystallization kinetics based on determining the time for 50%
recrystallized fractionand the Avrami exponentas a function of deformation conditions and steel composition is proposed for this ferritic stainless steel type 430 (Table 1).
- The effective annealing time for a fraction recrystallized is a critical input into the behavior model and, therefore, the correlation between measured and calculated volume fraction recrystallize as an output data over a wide range of processing conditions (Figures 7-11), is very satisfactory.
- The calculated results by this model are in good agreement with measured data. It can be used to investigate the static recrystallization and properties of ferritic stainless steelstype 430, and to provide the guidance to industry production.
- This integrated mathematical model has to be applied to get optimal range of the rolling parameters for the desired static recrystallization (mechanical properties).It
is also useful for the development of new steel grades and the development of optimized thermomechanical processing routes.
- The benefits of applying metallurgical modelling to the control of thermomechanical processing operations have been established particularly for flat product rolling. The current methodology, however, fails if applied to compute local microstructures and properties when the deformation conditions and the strain path change in complex ways during a pass.
To overcome this limitation, a new generation of models is required, which consider the microstructure on a finer scale. Hybrid modelling of the evolution of internal state variables,which describe the dislocation structures in the deformed material during and after each pass, and their effects on flow stress and recrystallization behaviour, provides a flexible methodology for through-process modelling.
- The results of the calculated recrystallized grain size and grain growth as well as the constitutive models based on semi empirical equations will be compared in the following publication to more sophisticated models based on cellular automata, vertex and Monte-Carlo-Potts methods.
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