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COMPARISON BETWEEN THE METHODS OF DETERMINING THE
CRITICAL STRESS FOR INITIATION OF DYNAMIC
RECRYSTALLIZATION IN 316 STAINLESS STEEL
M. Jafari, A. Najafizadeh
Department of Materials Engineering, Isfahan University of Technology, Isfahan, Iran.
mysmjafari@yahoo.com E-mail:
abbas.najafizadeh@gmail.com
Abstract
The critical stress for initiation of dynamic recrystallization (σc) could be identified by the
inflection point on the strain hardening rate (θ=dσ/dε) versus flow stress ( σ ) curve. Several
methods have been proposed to calculate this value on the basis of mathematical methods.
One of them was proposed by Stewart, Jonas and Montheillet in which this critical point
appears as a distinct minimum in the (-dθ/dσ versus σ ) through differentiating from θ
versus σ . Another one was presented by Najafizadeh and Jonas which obtain by modifying
the Poliak and Jonas method. According to this method, the strain hardening rate was plotted
against flow stress, and the value of σc was obtained numerically from the coefficients of the
third-order equation that was the best fit from the experimental θ - σ data.
In this study, hot compression tests were used in the range of 1000 to 1100 °C and strain
rates of 0.01-1s-1 and strain of 1 for the above mentioned methods to compare the value of the
σc for 316 stainless steel. The result shows that the method presented by Najafizadeh and
Jonas not only simpler than the previous one but also, it has a good agreement with
microstructures. Furthermore, the normalized critical stress for this steel was obtained uc =
σc/σp = 0.92.
Keywords: Dynamic Recrystallization; Critical Stress; Strain hardening rate; .
1. Introduction
Dynamic recrystallization (DRX) is a powerful tool for controlling microstructural
evolution during industrial hot deformation processing [1-5]. When this type of softening
mechanism is operating, both nucleation as well as growth take place while the strain is being
applied [1-2].
Calculation of the critical condition for the initiation of DRX is of considerable interest
for modeling of industrial processes [6-9]. It depends on the chemical composition of the
material, the grain size prior to deformation, and the deformation condition (T and ε& ) [7-10].
Several researchers have proposed mathematical relations to predict the initiation of DRX.
For example, M. R. Barnett, G. L. Kelly and P. D. Hodgson [11] have identified the critical
strain for initiation of DRX using the kinetics of static recrystallization (SRX) modified to
allow SRX to begin before the end of deformation. This approach defines the critical stress
for the initiation of DRX. On the basis of a dislocation density work hardening model, G.
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Gottstein, M. Frommert, M. Goerdeeler, and N. Schafer [12] have also predicted the critical
strain for initiation of DRX. The method developed by Ryan and McQueen [10] defines the
critical strain as the strain at which the experimental flow curve deviates from the idealized
dynamic recovery curve, based upon differences in the strain hardening behaviors associated
with DRX and dynamic recovery.
An alternative method based on the thermodynamics of irreversible processes was
developed by Poliak and Jonas [6-9]. This method eliminates the need for extrapolated flow
stress data; the onset of DRX can be identified by an inflection point in the θ - σ curve
corresponding to the appearance of an additional thermodynamic degree of freedom in the
system. Another one is Najafizadeh and Jonas [13] method which obtains by modifying the
Poliak and Jonas method. According to this method, the strain hardening rate was plotted
against flow stress, and the third-order equation that the best fit the experimental θ - σ data
from yield to the peak stress was found for each of the deformation conditions. Then the value
of εc was obtained numerically from the some coefficients of the third-order equation. In this
method the critical stress for the initiation of DRX is given by the simple relationship σc=B/3A, where A and B are two of the four coefficients of the third order equation.
In the present work, the two latter methods were compared according to microstructures
which obtained from quenched samples at critical points with different deformation
conditions.
The value of σc depends on the experimental conditions, if instead the normalized stressstrain curve is employed, according to the Najafizadeh and Jonas [13] method the normalized
stress is given by uc= σc/σp=-B'/3A', where A' and B' are two of four coefficients of
normalized third order equation. The value of uc is constant and independent of the
deformation conditions The validity of these relationships was examined for 316 stainless
steel.
2. Experimental procedure
The chemical composition of the 316 stainless steel employed in this work is given in
Table.1. This material was supplied with the form of a hot rolled bar with a diameter of 10
mm. Cylindrical samples 5 mm in diameter and 7.5 mm in height were prepares with their
axes aligned along the rolling direction.
For minimize the coefficient of friction during hot compression testing, glass plates were
put into graves (Rastegave) as lubricant. One-hit hot compression tests were carried out on a
dilatometer machine (BAHR Thermoanalysen GmbH) equipped with induction furnace and a
vacuum chamber.
The samples totally were deformed after preheated at 1150 °C for 15 min in a vacuum
chamber. Then they cooled to test temperature. Tests were performed at the temperature of
950-1100 °C with strain rates of 0.01-1s-1 to the strain of 1. Samples were cut along
compression direction and were polished electrochemically in a 65% nitric acid solution. In
order to minimize the coefficient of friction during hot compression testing, glass plates were
into graves as lubricant. In order to minimize the effect of friction on the stress-strain curves
the method of Ebrahimi and Najafizadeh [14] were used.
3. Results and Discussion
Fig. 1 present the typical DRX stress-strain curves of 316 stainless steel. In order to
calculate the critical stress for initiation of DRX (σc) the Poliak [6-9] and Najafizadeh [13]
methods were used.
The σc could be identified by the inflection point on the strain hardening rate (θ=dσ/dε)
versus flow stress ( σ ) curve. The Poliak method requires differentiation of the stress- strain
curve, but short- range noise can render such differentiation calculus impossible. The
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variation in the derivate of the true stress can be several orders of magnitude larger than the
mean value. In order to solve this problem, a smoothed stress-strain curve was obtained by
modeling the region of interest (the region of plastic deformation that encompasses the stress
peak) as a 9th order polynomial. These polynomials are presented as common form of below.
σ = a + bε + cε 2 + dε 3 + eε 4 + fε 5 + gε 6 + hε 7 + iε 8 + jε 9
(2)
The 9th order polynomials were obtained for the curves of Fig. 1 is presented in the table
2.
As differentiation of the polynominal is simple and free of noise, the work hardening
profile can be readily derived. The θ-σ plot corresponding to the stress-strain curve of 1100
°C with the strain rate of 0.1 s-1 is shown in Fig. 2.
The θ-σ curve is directly related to concave-downward shape of the stress-strain curve. In
the absence of dynamic recrystallization, the work hardening rate will taper off to zero as the
strain goes to arbitrarily large values. However, dynamic recrystallization causes a downward
inflection in the θ-σ curve, leading to zero and negative work hardening rates (corresponding
to the peak and post peak softening, respectively).
The method involves differentiating the above θ-σ curve. The critical point then appears
as a distinct minimum represents the initiation of dynamic recrystallization (Fig.3).
2- Najafizadeh and Jonas method
In the approach of Poliak and Jonas [6-9, 15] and N. D. Ryan and H. J. McQueen [10], the
initiation of DRX is associated with the point of inflection in the θ-σ curve. To plot this kind
of curve, it is necessary to find an equation that fits the experimental θ-σ data from yield until
the peak stress. The simplest equation that has an inflection point is:
θ = Aσ 3 + Bσ 2 + C σ + D
(3)
Where:
A, B, C, D is constants for a given set of deformation conditions. All the third order
equations obtained from this study are presented in Table.3. Differentiation of equation 3
with respect to σ results in:
dθ
= 3 Aσ 2 + 2 Bσ + C
dσ
d 2θ
= 0 ⇒ 6 Aσ + 2 B = 0
dσ 2
σc = −
B
3A
(4)
Fig.4 shows for example a θ-σ curve from yield until the peak stress at 1100 °C and strain
rates of 0.1 s-1. In the table 4 the compressions of these two methods are presented and the
fraction's values σc/σP in a certain range show approximately independence to various
deformation conditions. As can be seen in Fig.5 those obtained from Najafizadeh and Jonas
method are in good agreement with microstructures from quenched samples at critical points
in different condition. Thus, this method is simpler than another one. So we suggest usage of
this method.
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The normalized stress-strain curve
Once normalized stress (u=σ/σp) - stain (w=ε/εp) plot have been established, the flow
stress at any strain, strain rate and temperature can be calculated from this single curve. Some
normalized stress-strain curves of the 316L stainless steel deformed under different conditions
of deformation are illustrated in Fig.6. The normalized strain hardening rate (θN=du/dw)
versus normalized stress (u) plots obtained from such curves have inflection points that
present the onset of initiation of DRX. For a given steel, the inflection points of θN-u curve
correspond to the approximately constant normalized critical stress uc= σc/ σp = 0.92.
Consequently for a given steel deformed under different conditions of deformation the ratio of
the critical strain to the peak strain, wc=εc/εp, is also approximately constant.
To obtain the critical stress for initiation of DRX under different conditions of
deformation, the normalized stress-strain curve was used. The best fit equation corresponding
to these data had the form:
θ N = A′u 3 + B ′u 2 + C ′u + D ′
(5)
The coefficients of this equation do not depend on the deformation conditions. The critical
stress for initiation of DRX can be calculated:
dθ N
= 3 A′u 2 + 2 B ′u + C ′
du
(6)
d 2θ N
= 0 ⇒ 6 A′u c + 2 B ′ = 0
du 2
uc = −
B′
3 A′
(7)
Using equation 7, the critical stress for any deformation conditions can be calculated by
using σp for such condition. This approach also helps to explain the constancy of the σc/σp.
As can be seen in Fig. 6 there are small discrepancies that arise from a weak dependence of
the curves on the value of Z. This source of error can be reduced by deriving 4 normalized
curves pertaining to four different sets of deformation conditions and then averaging the third
order of equation 6 (Fig.7):
θ = −375.55σ 3 + 1038 .2σ 2 − 959.52σ + 296.83 (8)
Thus the uc will be;
uc =
− 1038.2
= 0.92 (9)
3 × 375.55
The uc value is expected to be constant under all conditions of deformation. This
hypothesis can be assessed by plotting the measured values of σc versus those calculated from
equation 9. Such a comparison is presented in Fig. 8. As can be seen in this figure there is a
reasonably good agreement between these two sets of values.
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4. Conclusion
The principal conclusions that can be drawn from the present work are the following:
1. The Najafizadeh et.al. method for calculated the onset of DRX is in good agreement with
microstructures obtained from quenched samples at this point for different condition of
deformation.
2. The method of Najafizadeh et al. is simpler than Poliak et.al. because he used the third in
place of nine order equation.
3. The normalized critical stress for initiation of DRX (uc) that calculated from the each
method is approximately constant and independent of the deformation conditions.
4. The critical value of uc for 316L stainless steel obtained from Najafizadeh et.al. method is
0.92.
Acknowledgments
The author expresses theirs thanks to the Center of Excellence for Steel for supporting of
this work
References
1. J. J. Jonas, Materials science and Engineering, 1994, A184, pp.155.
2. R. D. Doherty, D. A. Hughes, F.J. Humphreys, J. J. Jonas, D. Juul Jensen, M. E. Kassner,
W.E. King, T. R. McNelley, H. J. McQueen, A. D. Rollet, Materials Science and
Engineering, 1997, A238, pp.219.
3. H. J. McQueen, S. Yue, N. D. Ryan and E. Fry, Materials processing Technology, 1995,
A53, pp.293.
4. G. R. Stewart, J. J. Jonas and F. Montheillet, ISIJ International, 2004, Vol. 44, No.9,
pp.1581.
5. F. Montheillet, J. J. Jonas, Encyclopedia of Applied Physics, 1996, Vol. 16, pp.205.
6. E. I. Poliak, J. J. Jonas, Acta mater, 1996, Vol. 44, No.1, pp.127.
7. E. I. Poliak, J. J. Jonas, ISIJ International, 2003, Vol. 43, No.5 , pp.684.
8. E. I. Poliak, J. J. Jonas, ISIJ International, 2003, Vol. 43, No.5 , pp.692.
9. J. J. Jonas, E. I. Poliak, Mater. Sci. Forum, Acta mater, 2003, Vols. 426-432, pp.57.
10. N.D. Ryan and H. J. McQueen, Can, Metall, Q., 1990, Vol.29, pp.147
11. M. R. Barnett, G. L. Kelly, P. D. Hodgson, Scripta mater, 2000, Vol. 43, pp. 365.
12. G. Gottstein. M. Frommert, M. Goerdeeler, N. Schafer, science and Engineering, 2004,
A387-389, pp.604.
13. A. Najafizadeh , J. J. Jonas, ISIJ International, 2006, Vol. 46, No. 11, pp. 1679.
14. R. Ebrahimi and A. Najafizadeh, "A new method for evaluation of friction in bulk metal
forming", Materials processing Technology, Vol. 152, Issue 2, 2004, PP.136-143.
15. E. I. Poliak, J. J. Jonas, ISIJ International, 2004, Vol. 44, No.11 , pp.1874.
Table.1. Chemical composition of 316 stainless steel
%Cr %Mo %Cu
%C
%Si
%S
%P %Mn %Ni
0.043 0.285 0.022< 0.044 1.635 10.225 16.679 2.207 0.489
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Table.2. Coefficients for stress (σ) versus strain (ε) equations T=1000-1100°C
ε& =0.01-1 s-1.
Strain rate
σ −ε
T°°C
(s-1)
σ = 22.95 + 1.2 × 103 ε − 2.3 × 10 4 ε 2 + 1.4 × 105 ε 3 − 5.1 × 105 ε 4 +
1
1100
1.14 × 106 ε 5 − 1.6 × 106 ε 6 + 1.38 × 106 ε 7 − 6.57 × 105 ε 8 + 1.33 × 105 ε 9
σ = 24 + 1.84 × 10 3 ε − 2.07 × 10 4 ε 2 + 1.29 × 10 5 ε 3 − 4.71 × 10 5 ε 4 +
0.1
1.07 × 10 6 ε 5 − 1.5 × 10 6 ε 6 + 1.3 × 10 6 ε 7 − 6.2 × 10 5 ε 8 + 12.6 × 10 4 ε 9
σ = 18.23 + 1.34 × 103 ε − 1.6 × 104 ε 2 + 1.1 × 105 ε 3 − 4.3 × 105 ε 4 +
0.01
1000
and
1.03 × 106 ε 5 − 1.51× 106 ε 6 + 1.34 × 106 ε 7 − 6.54 × 105 ε 8 + 1.36 × 105 ε 9
σ = 22 + 2.4 × 103 ε − 2.3 × 104 ε 2 + 1.3 × 105 ε 3 − 4.56 × 105 ε 4 +
0.1
9.88 × 105 ε 5 − 1.35 × 106 ε 6 + 1.13 × 106 ε 7 − 5.28 × 105 ε 8 + 1.05 × 105 ε 9
Table.3. Coefficients for stress (θ) versus strain (σ) equations T=1000-1100°C and
ε& =0.01-1 s-1 obtained from yield until the peak stress.
T °C
Strain rate
(S-1)
θ/σrelation
1
θ =-0.21σ3 +61.63σ2 -5991σ +194146
1100
1000
0.1
θ =-0.018σ3 +6.9σ2 -874.4σ +37071
0.01
θ =-0.32σ3 +63.67σ2 -4168.7σ +91045
0.1
θ =-0.012σ3 +5.17σ2 -765.99σ +38068
Table.4. Two sets of results obtained from two kinds of methods mentioned in this
article.
Poliak-Jonas method
Najafizadeh-Jonas method
Deformation condition
1100 °C، 0.1 s-1
σc
σc/σp
σc
σc/σp
98.42
0.95
97.82
0.94
124.23
0.88
127.77
0.90
65.84
0.95
66.32
0.96
147.9
0.91
143.61
0.88
-1
1100 °C، 1 s
1100 °C، 0.01 s-1
-1
1000 °C01، 0.1 s
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Fig.1. Stress- strain curves obtained from one-hit compression tests
2000.00
θ=dσ/dε
1500.00
1000.00
500.00
0.00
-500.00
0
20
40
60
80
100
120
σ
Fig. 2. The θ-σ plot corresponding to the stress-strain curve at 1100 °C and strain rate
0.1 s-1
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Fig.3. The dθ/dσ versus σ curve at 1100 °C and strain rates of 0.1 s-1, these graphs
must be centered on the inflection point to determine the critical strain for
onset of DRX.
Fig.4. The θ-σ plot corresponding to the stress-strain curve at 1100°C and strain rate
0.1 s-1 obtained from yield until the peak stress.
Fig.5. Microstructures obtained from quenched samples at critical strian for initiation
of DRX in different condition. ; Arrow showed bulging and serration produce
by the initiation of DRX.
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Fig.6. The normalized stress -stain curves obtained under different conditions of
deformation.
Fig.7. Normalized strain hardening rate versus normalized stress at 1100°C
and strain rates of 0.1 s-1.
Fig.8. Critical stress for the initiation of DRX (σc). Here the measured values are
compared with those estimated using the relation σ= 0.92σP.