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Acta Metallurgica Slovaca, 11, 2005, 3 (362 - 368) 362
STATISTICAL COMPARISON OF DIFFERENT METHODS OF CALCULATION
OF PARAMETERS DESCRIBING DYNAMIC RECRYSTALLIZATION
Kubina T. 1 , Schindler I. 1 , Heger M. 1 , Plura J. 1 , Bořuta J. 2 , Dänemark J. 3 , Hadasik E. 4
1 VŠB - Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech Republic,
tomas.kubina@vsb.cz
2 VÍTKOVICE - Research & Development, s.r.o., Pohraniční 31, 706 02 Ostrava, Czech
Republic, josef.boruta@vitkovice.cz
3 Třinecké železárny, a. s., Průmyslová 1000, 739 70 Třinec, Czech Republic,
janusz.danemark@trz.cz
4 Politechnika Śląska, Krasińskiego 8, 40-019 Katowice, Poland, eugeniusz.hadasik@polsl.pl
STATISTICKÉ SROVNÁNÍ ROZDÍLNÝCH POSTUPŮ PŘI VÝPOČTU PARAMETRŮ
POPISUJÍCÍCH DYNAMICKOU REKRYSTALIZACI
Kubina T. 1 , Schindler I. 1 , Heger M. 1 , Plura J. 1 , Bořuta J. 2 , Dänemark J. 3 , Hadasik E. 4
1 VŠB – Technická univerzita Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech Republic,
tomas.kubina@vsb.cz
2 VÍTKOVICE - Výzkum a vývoj, s.r.o., Pohraniční 31, 706 02 Ostrava, Česká Republika,
josef.boruta@vitkovice.cz
3 Třinecké železárny, a. s., Průmyslová 1000, 739 70 Třinec, Česká Republika,
janusz.danemark@trz.cz
43 Politechnika Śląska, Krasińskiego 8, 40-019 Katowice, Polsko, eugeniusz.hadasik@polsl.pl
Abstrakt
V současné době program ENERGY 4.0, vyvinutý pro výpočet parametrů
popisujících dynamickou rekrystalizaci (DRX), v sobě zahrnuje dvě metodiky výpočtu. Vstupní
data pocházející z torzního plastometru SETARAM – VÍTKOVICE jsou nejdřív zpracovány
konvenční metodou postupných lineárních regresí. V této fázi jsou vyloučeny teplotní hladiny
při kterých není strukturní stav zkoušené oceli jednofázový. Následující automatický výpočet,
využívající simplexní algoritmus nelineární regrese a, může jednotlivé parametry vyjádřit
mnohem přesněji. Tento výpočet je více objektivnější a dosahuje menší statistické chyby,
výsledné parametry ale ne vždy odpovídají fyzikálním předpokladům. V celkovém množství 22
ocelí různého chemického složení od ocelí jednoduchého strukturního stavu přes mikrolegované
oceli různých typů až k vysocelegovaným ocelím a korozivzdorným austenitickým ocelím.
Klíčovým kritériem pro zařazení do vyhodnocované skupiny ocelí bylo zachování jednofázové
struktury v celé oblasti zkušebních teplot. Máme tak možnost porovnat vliv obou metod na
vypočtené parametry a taktéž přesnost obou metod na určení hodnot maximálního napětí a
deformace v píku.
Abstract
Program ENERGY 4.0, developed for calculation of parameters describing dynamic
recrystallization (DRX), includes now two ways of methodology of calculation. Input data
coming from the torsion plastometer SETARAM - VÍTKOVICE is first processed by a
conventional method of successive linear regressions. In this stage the exclusion of temperature

Acta Metallurgica Slovaca, 11, 2005, 3 (362 - 368) 363
levels is carried out, in which a state featured by one phase of tested steels is not maintained.
The following automatic computation using the simplex algorithm of non-linear regression can
make particular parameters for a description of DRX more precise. Those calculations are
entirely objective and yield in the least statistical errors but their results can not always match up
the physical-based idea. In total 22 steels of different chemical composition - from simple
structural steels, through microalloyed steels of various types, high alloyed tool steels up to
corrosion-resisting (stainless) austenitic steels - were included in the analyzed data set. A key
(decisive) criterion for including in an evaluated group of steels was maintaining of mono-phase
structural state in the whole range of evaluated temperatures of testing. One can compare
influence of both methods on calculated parameters, as well as accuracy of both methods in
determination of values of the maximum stress and peak to strain.
Key words: Activation energy, hot forming, steel, dynamic recrystallization
1. Fundamental theory for description of actiavtion energy
Activation energy Q [J·mol -1 ] is an important material constant dependent on the
chemical composition and microstructure of the hot formed material, deformation behaviour of
which is largely influenced by this variable. Knowledge of its value is very useful, e.g. for
description of dynamic recrystallization kinetics as well as stress-strain curves in continuous
deformation. For finding out the Q-value, it is suitable to use a solution of modified equation [1]:
⎡ − Q ⎤
e & = C. exp⎢
.[ sinh( α.
σ )] n
max
R.
T
⎥
(1)
⎣ ⎦
where ė [s -1 ] is strain rate; C [s -1 ], a [MPa -1 ] and n are other material constants, T [K] is
deformation temperature, R = 8.314 Jmol -1·K -1 , σ max [MPa] is deformation resistance associated
with the peak stress. The traditional way of assessing the constants in Eq. 1 was given e.g. in [1].
The principle lies in the application of specific characteristics of the hyperbolic sine function.
We suppose that Eq. 1 can be converted to the relationship
⎡ − Q ⎤ n
e& = C1 . exp⎢
. σ max
R.
T ⎥ (2)
⎣ ⎦
for low stress values (i.e. a particular high temperature level), or to the relationship
⎡ − Q ⎤
e& = C2 . exp⎢
.exp( β.
σ max )
.
⎥
(3)
⎣ R T ⎦
for high stress values (i.e. a selected low temperature level). The constants n and β can be gained
when applying Eqs. 2 and 3, respectively. The key constant β enables to find the α-value,
assuming that α = β / n.
2. Calculation DRX parameters by software ENERGY 4.0
2.1 System of Linear Regressions
The above mentioned equations can be solved by a graphic method using multiple
linear regression analysis. The computing program ENERGY 4.0 was developed (in language

Acta Metallurgica Slovaca, 11, 2005, 3 (362 - 368) 364
Delphi) [2,3] which has made more sophisticated work with the input data set possible. Such
data table can be obtained from continuous torsion tests. In every row of that table, the
calculated peak stress σ max [MPa] and strain to peak e p are given, depending on temperature and
strain rate of individual torsion tests. Initially the experimental data is processed step-by-step
using partial linear regressions after plotting the selected points in proper co-ordinates:
• first the n-value is obtained as a slope of the regression line in co-ordinates ln σ - ln ė (a
high temperature level);
• then the β-value is determined in coordinates σ - ln ė (a low temperature level);
• after calculation of the α-value, the final regression in complex co-ordinates 1/T – (ln ė
- n·sinh(α·σ)) is realized utilizing all experimental data, and thus the constants Q and C
can be obtained. Principles of work with the ENERGY 4.0 are described in work [4].
2.2 Refining Calculations Using the Simplex Algorithm
After elimination of the unfitting data, software Energy 4.0 enables to refine the
calculations described above. Constants in Eq. 1 are determined by a non-linear regression
considering two mutually independent variables - temperature and strain rate. The nonderivation
methods of searching the extreme of proper function exhibited the best stability as
well as agility of computation. Thus the simplex algorithm was adopted which gave results (i.e.
values Q, C, a and n) in a few seconds [3]. Those calculations are entirely objective and yield in
the least statistical errors but their results can not always match up the physical-based idea.
Discussion of the results from both ways of calculation is included in part 3.
2.3 Kinetics of Dynamic Recrystallization
Peak strain e p is approximately linked up with the initiation of dynamic
recrystallization. The e p -value depends very simply on the Zener-Hollomon's parameter Z [s -1 ]
[5,6]:
e
p
w
= u ⋅ Z
(4)
The parameter Z involves the Q-value that represents the material factor (chemical
composition, structure, etc.). It can be defined as the temperature-corrected strain rate [7]:
⎛ Q ⎞
Z = é ⋅exp ⎜ ⎟
(5)
⎝ R ⋅T
⎠
where T [K] is deformation temperature.
ENERGY 4.0 using linear regression of Eq. 4 in logarithmic formulation computes
constants u and w. These are determined in either case of previous calculations.
3. Statistic evaluation of both computation methods
Input data for calculation of kinematics of the dynamic recrystallization was measured
on the torsion plastometer SETARAM. In total 22 steels of different chemical composition -
from simple structural steels, through microalloyed steels of various types, high alloyed tool
steels up to corrosion-resisting (stainless) austenitic steels - were included in the analyzed data
set. Chemical composition of these steels is given in Tab. 1. A key (decisive) criterion for

Acta Metallurgica Slovaca, 11, 2005, 3 (362 - 368) 365
including in an evaluated group of steels was maintaining of mono-phase structural state in the
whole range of evaluated temperatures of testing.
Table 1 Target chemical composition of tested steels in wt. %
11375 0.10 C – 0.29 Mn – 0.15 Si – 0.05 Al
11523 0.18 C – 1.10 Mn – 0.54 Si – 0.05 Al
Free-Cutting 0.13 C – 1.45Mn – 0.03 Si – 0.30 S
X52
0.10 C – 1.25 Mn – 0.35 Si – 0.02 Nb
X60
0.11 C – 1.36 Mn – 0.29 Si – 0.04 Nb
X70
0.10 C – 1.40 Mn – 0.39 Si – 0.04 V – 0.02 Nb
X80
0.09 C – 2.04 Mn – 0.57 Si – 0.23 Mo – 0.04 V – 0.03 Nb
S700MC 0.12 C – 1.76 Mn – 0.53 Si – 0.14 V – 0.03 Nb – 0.003 B
WB36
0.16 C – 1.10 Mn – 0.40 Si – 0.52 Cu – 1.18 Ni – 0.21 Cr – 0.36 Mo – 0.03 Nb
11MnSi6 0.084 C – 1.40 Mn – 0.86 Si
23MnB4 0.22 C – 0.91 Mn – 0.07 Si – 0.28 Cr – 0.003 B
14209 0.98 C – 1.04 Mn – 0.51 Si – 1.49 Cr
15260 0.50 C – 0.80 Mn – 0.35 Si – 1.07 Cr
SA387GR11 0.17C – 0.57 Mn – 0.53 Si – 0.36 Ni – 1.44 Cr – 0.59 Mo
SiCrMoCu 0.23 C – 1.06 Mn – 1.00 Si – 0.16 Cu – 1.31 Cr – 0.31 Mo
Superferrite 0.01 C – 0.51 Mn – 0.49 Si – 0.50 Ni –17.42 Cr – 3.8 Mo – 0.03 V – 0.07 Al – 0.07 Nb
CrNi
0.06 C – 0.69 Mn – 0.47 Si – 9.7 Ni – 17.2 Cr – 0.57 Mo
CrMn
0.48 C – 16.6 Mn – 0.57 Si – 0.27 Ni – 15.4 Cr
5H17G17 0.50 C – 17.1 Mn – 0.57 Si – 16.7 Cr
CrNiMo 0.02C – 0.67 Mn – 0.52 Si – 15.0 Ni – 18.1 Cr – 5.9 Mo
CrNiMoN 0.03 C – 1.56 Mn – 0.52 Si – 5.35 Ni – 22.4 Cr – 2.7 Mo – 0.06 V –0.192 N
19830 0.81 C – 0.23 Mn – 0.27 Si – 0.25 Ni – 4.03 Cr – 4.4 Mo – 2.03 V –5.37 W – 1.01 Co
3.1 Calculated parameters describing DRX
The first of statistic results is a mean deviation ∆ for activation energy Q defined as
follows:
m
∑
M
A
Qi
− Qi
i=
1
∆Q
=
(6)
m
where upper index M means a value calculated by means of a system of linear regressions, index
A is used for designation of the automatic simplex method of calculation. The results for specific
parameters are summarized in Tab. 2.
Table 2 Summary of statistic results
Parameter ∆ T-test Correlation
Mean value P-value
Correlation
coefficient r
P-value
Q -29.467 0.0020 0.9641 0.0000
α -0.524 0.2780 0.5600 0.0082
n 0.00074 0.0047 0.669 0.0010
logC -1.077 0.0019 0.906 0.0000
u 0.0006 0.0290 0.999 0.0000
w 0.0085 0.00002 0.995 0.0000
Further t-test was carried out for the case of a difference between dual values. Here
statistical significance of the difference between values gained by linear regression and

Acta Metallurgica Slovaca, 11, 2005, 3 (362 - 368) 366
automatic computation was monitored. It was tested whether the difference between these values
equals zero for significance level α = 0,05. The less is the achieved significance level, so called
P-value, the more unmaintainable, i.e. less credible, is the null hypothesis. The last test consisted
in determination of correlation between both ways of calculation, again on the significance level
α = 0,05. The values of correlation coefficient and P-value are shown in Tab. 2.
From the values in Tab. 2 and a graphic expression of relations between values
obtained for both methods of calculation (see Fig. 1) we can define the following conclusions.
The activation energy Q has always been evaluated by the automatic simplex method in smaller
values. The same is given for parameter α and a logarithmic value of coefficient C. From the t-
test follows that a difference between parameters is statistically significant in the case of all
parameters monitored, except for parameter α. More significant is here a comparison of
correlation between both computational methods. In case of parameter α and n the correlation is
less significant, which results from a limitation of the method that uses a system of linear
equations where these parameters are numbered for selected temperature levels and affected by a
small number of points used for calculation. High values of the correlation coefficient in case of
parameters u and w can be attributed to the fact that the calculation is based on values of
activation energy Q which sufficiently correlate between each other.
Fig.1 Relationships between constants Q, n, a, C calculated by the multiple linear regression (upper index M) and nonlinear
simplex method (upper index A)

Acta Metallurgica Slovaca, 11, 2005, 3 (362 - 368) 367
3.2 Comparison between measured and calculated values σ max and e p
Relative deviations σ M max and σ A max from measured values of maximum stress of
deformation σ max are positive and negative for both ways of calculation, their distribution is
approximately symmetrical. In Fig. 2 is shown that ion case of automatic calculation the relative
deviations from the value σ max are shifted to negative values. The arithmetic mean for relative
deviations between the maximum stress of deformation calculated manually and the measured
maximum stress of deformation equals 0.104 %.
Correlation between measured and calculated values of the maximum stress is better
for the automatic way of calculation (correlation coefficient r = 0.9969) than for the manual way
of calculation (r = 0.9956). The same situation is in the case of variance (S M = 19369 against
S A = 13343).
When similar results for a size of strain to peak are compared, it is found out that the
distribution of relative deviations for both methods of calculation is approximately symmetrical
(see Fig. 3). The correlation between calculated and measured values of strain to peak is in both
cases less significant than for the maximum stress. Correlation coefficient r = 0.9714 is exactly
the same for both methods of calculation. The variance values are only slightly different, for
manual calculation a value of S M = 1.2098, for automatic computation S A = 1.2118.
Fig.2 Histograms for distribution rate of relative
deviations of maximum stress for manual (M)
and automatic (A) calculation
Fig.3 Histograms for distribution rate of relative
deviations of strain to peak for manual (M)
and automatic (A) calculation
4. Conclusions
Software ENERGY 4.0 developed in environment Delphi makes it possible to
calculate the activation energy in hot forming, together with other parameters necessary for
description of maximum stress of deformation and strain to peak necessary for the start of
dynamic recrystallization by means of the given equations.
Parameters defining DRX were calculated in two ways, based on the same set of input
data. A conventional way of the multiple linear regression in combination with interactive work
in ENERGY 4.0 enabled that temperature regions with signs of phase transformations could be

Acta Metallurgica Slovaca, 11, 2005, 3 (362 - 368) 368
eliminated. The non-linear regression using simplex algorithm is very fast and gives more
accurate results for the whole group of evaluated steels [8]. It is more advantageous as far as
numbering of constants for calculation of maximum deformation resistance is concerned. As for
accuracy of the strain to peak determination, both methods are comparable.
The difference between methods is in approach to the parameters entering the
equations. In case of use of the multiple linear regression the values of activation energy are in
the foreseen conventional limits. When the non-linear regression is used, the activation energy
for steel grades 23MnB4, 11523, 14209 is even below the value of activation energy for autodiffusion
of iron 270 kJ.mol-1. Then the activation energy loses its physical essence [9,10] and it
becomes only one of material constants. However, the result consists in a desirable decreased
difference between measured and predicted data [e p ;σ max ].
Acknowledgement
This work was carried out under project number MSM 6198910015, funded by the
Ministry of Education, Youth and Sports of the Czech Republic.
Literature
[1] Sellars C. M., Tegart W. J. McG.: International Metallurgical Review, 1972, No. 158, p. 1.
[2] Kopetschke I., Schindler I.: Transactions of the Institute of Mining and Metallurgy of
Ostrava, Metallurgical Series, 1991, No. 1, p. 229.
[3] Podstuvka D.: Diploma thesis. VŠB - Technical University of Ostrava, Faculty of
Metallurgy and Materials Engineering, 2003.
[4] Schindler I., Podstuvka D., Bořuta J.: In: CO-MAT-TECH 2003, Trnava 2003, p. 925.
[5] Kliber J. et al.: Steel research, 1989, No. 11, p. 503.
[6] Schindler I., Kliber J., Bořuta J.: In: METAL '94. Ostrava 1994, p. 132.
[7] Treatise on Materials Science and Technology. Vol. 6 - Plastic Deformation of Materials.
Ed. Arsenault R. J., Academic Press, New York 1975.
[8] Kubina T. et al: In: FORMING 2005. Lednice 2005, p. 145.
[9] Kliber J., Schindler I., Bořuta J.: In: PLAST '96. Ustroń 1996, p. 17.
[10] Gronostajski Z. J.: ibid., p. 23.