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Characteristic points of stress–strain curve at high temperature

Authors:
Ramin Ebrahimi at Shiraz University
Ramin Ebrahimi
Soheil Solhjoo at University of Groningen
Soheil Solhjoo
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Determination of critical points on hot stress-strain curve of metals is crucial in thermo-mechanical processes design. In this investigation a mathematical modeling is given to illustrate the behavior of metal during hot deformation processes such as hot rolling. The critical strain for the onset of dynamic recrystallization has been obtained as a function of strain at the maximum stress. In addition, the transition strain from static recrystallization to full metadynamic recrystallization has been presented to form an equation as a function of peak strain, peak stress and steady-state stress. The results of this mathematical modeling are in a good agreement with the experimental data.
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International Journal of ISSI, Vol.4 (2007), No. 1,2, pp. 24-27
Characteristic Points of Stress-Strain Curve at High Temperature
R. Ebrahimi1* and S. Solhjoo2
Department of Materials Science and Engineering, School of Engineering, Shiraz University, Iran
Abstract
Determination of critical points on hot stress-strain curve of metals is crucial in thermo-mechanical
processes design. In this investigation a mathematical modeling is given to illustrate the behavior of metal during
hot deformation processes such as hot rolling. The critical strain for the onset of dynamic recrystallization has
been obtained as a function of strain at the maximum stress. In addition, the transition strain from static
recrystallization to full metadynamic recrystallization has been presented to form an equation as a function of
peak strain, peak stress and steady-state stress. The results of this mathematical modeling are in a good
agreement with the experimental data.
Keywords: Dynamic Recrystallization, Metadynamic Recrystallization, Transition Strain, Critical Strain.
Introduction
Evaluation of forces and forming energy is the
most important aspect of mechanical design in metal
forming processes 1,2). This requires a knowledge of
the flow stress of metals. Predicting and controlling
the microstructure during hot deformation in
industrial processes such as hot rolling is of great
importance. Microstructure evolution and
deformation mechanisms during deformation are
closely related to flow stress 3-7). Determination of
mechanisms and microstructural evolution by
methods such as optical microscopy, scanning
electron microscopy (SEM) and transmission
electron microscopy (TEM) is possible but expensive
and time consuming. Many researchers have tried to
predict microstructural evolution from stress-strain
curves; thus, by proper designing of thermo-
mechanical processes, the suitable microstructure is
formed.
Flow stress is a function of dislocation density
(
ρ
) 8), and dislocation density at high temperature
is a function of some parameters such as primary
microstructure, temperature and strain rate.
Therefore, for a given microstructure, temperature
and strain rate condition, the change of stress versus
strain is closely related to metal microstructural
evolution. Appearance of the maximum stress on the
stress-strain curve and its slow decrement to the
steady-state stress are the characteristics of dynamic
recrystallization (DRX). This phenomenon usually
occurs in the metals with low to medium stacking
fault energy (SFE) such as Iron, Copper, Austenitic
Stainless Steel, etc.
*Corresponding author:
Tel: +98-711-6286531 Fax: +98-711-6287294
E-mail: ebrahimy@shirazu.ac.ir
Address: Dept. of Materials Science and Engineering,
School of Engineering, Shiraz University, Iran
Work hardening and dynamic recovery (DRV)
mechanisms control the flow stress up to the
maximum stress. DRX occurs before the maximum
stress, but since the latter is easier to determine, it is
practically used in the design of thermo-mechanical
processes even though many researches have tried to
locate the exact point of the on set of DRX on the
stress-strain curve, i.e. the critical strain 9-12).
After the peak of stress-strain curve is reached,
the rate of softening increases to a maximum value at
a particular strain. This particular strain is called
"transition strain to full metadynamic
recrystallization" 13,14). Determining this transition
strain is very important in industrial processes such
as hot rolling. This means that the process must be
designed in such a way to allow the nucleation of
new grains during deformation. Afterwards, full
metadynamic recrystallization happens in the time
between rolling stands. Therefore, critical points are
essential in the design of thermo-mechanincal
processes. The purpose of this paper is to present a
mathematical model which shows the metal
behaviour during hot deformation. This model can
exactly determine the critical strain, εC, for the onset
of DRX and the transition strain, εT, for static-
metadynamic recrystallization transition into full
metadynamic recrystallization after deformation.
Mathematical analysis of flow stress
Flow curves of metals having low to medium
SFE show clear DRX behavior. Thus, these curves
always have only one peak. McQueen et al. have
modeled the stress-strain curve up to the peak by 9,15):
)1(1exp
C
PPP
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
ε
ε
ε
ε
σ
σ
1. Assistant Professor
2. B. S. Student
International Journal of ISSI, Vol.4 (2007), No. 1,2
25
where C is a constant and must be determined for
each metal and σP and εP are the maximum stress and
strain at the maximum stress, respectively.
Figure 1 is a schematic hot flow curve when
DRX occurs. This curve indicates two zones. In zone
I, work hardening prevails DRV, so the flow stress
increases up to a maximum value. In zone II, DRV
and DRX occur simultaneously, thus flow stress
gradually decreases and at large strains reaches a
steady-state value. The derivative of flow stress is
positive in zone I and negative in zone II, and
approaches zero in the steady-state zone. Equation 2
demonstrates the stress-strain relationship after the
maximum stress as a complement for McQueen
equation 16):
()
)2(
22
exp
2
1⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−−−+=
P
P
SPS C
ε
εε
εσσσσ
where σP is the maximum stress, εP is the strain at the
maximum stress and σS is the steady-state stress. In
order to determine the constant, C1, a point, K is
chosen on the curve such that K<1 and σK>σS. Then
C1 is given by equation 3:
()
)3(ln
12
2
2
1⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
+−
=
SK
SP
P
KK
C
σσ
σσ
ε
where σK is calculated by Eq. 1 at εK = KεP.
The equations represented by this model require
the values of stress and strain at the peak and stress at
the steady-state zone. These parameters can be
calculated by kinetic equations 16).
Fig. 1. Schematic flow curve at high temperature for
DRX.
Critical strain for the onset of DRX
During the deformation, an increase in the work
hardening increases the dislocation density leading to
a critical microstructural condition at which new
grains nucleate and new high-angle boundaries grow.
This phenomenon is called dynamic recrystallization,
DRX. Gradually, the dislocation density increases in
other areas as well. Therefore, the flow stress
increases up to a maximum value and the rate of
softening mechanism prevails work hardening
afterwards. Thus, flow stress reaches a steady-state
condition at which the rate of generation and
annihilation of dislocations become equal each other.
The critical strain at the onset of DRX can be
determined metallorgraphically from the observation
of microstructure of quenched specimens. This
technique requires a large number of specimens
deformed to different strains. On the other hand, the
critical strain thus obtained is not precise 11). Ryan
and McQueen observed an inflection in the
ε
σ
θ
∂
∂
=
vs. σ curve 15). Then Poliak and Jonas showed that
this inflection corresponds to the initiation of DRX
and occurs at εC=0.55εP for Nb-microalloyed steels
11). This is when the critical strain of the onset of
DRX, formerly reported by microscopic observation,
was εC=0.8εP 12).
In this research, applying Ryan and McQueen
theory and mathematical calculations, an equation is
introduced to calculate the critical strain at the onset
of DRX. The derivative of equation 1 gives:
)4(
11 ⎥
⎦
⎤
⎢
⎣
⎡−=
P
C
d
d
εε
σ
ε
σ
Fig. 2. Derivative of stress with respect to strain vs.
stress.
Fig. 2 is obtained from Eq. 4 by assuming
εP=0.85 and C=0.2 just as a sample data to show the
form of Eq. 4. The schematic shape of this curve is
exactly like the ones reported in the references, with
only one inflection 11). As Figure 3 and Figure 4
show, the second derivative of
ε
σ
θ
∂
∂
= with
respect to σ must be zero, in order to determine the
critical point.
)5(0
2
2
=
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
σ
ε
σ
d
d
by solving Eq. 5 we get the critical strain for the
onset of DRX as a function of strain at the maximum
stress:
)6(
11
1PC C
C
εε
⎥
⎦
⎤
⎢
⎣
⎡−−
−=
0
40
80
120
160
012345
Strain
Stress (MPa)
WH
DRV
WH-DRV-DRX
Steady State
DRX
region (I)
region (II)
International Journal of ISSI, Vol.4 (2007), No. 1,2
26
where C is the constant of equation 1 and varies
between 0.1 to 0.3 for different alloys 15). Therefore,
the critical strain values vary between 0.46εP to
0.49εP for different alloys which is in a good
agreement with values (εC=0.55εP) reported by other
researches for Nb-microalloyed steel 11). As a result,
if the flow stress equation for different alloys is
available, the exact amount of the critical strain for
the onset of DRX could be calculated by Eq. 6.
Fig.3. Derivative of work hardening rate with respect
to stress vs. stress.
Fig. 4. Second derivative of work hardening rate with
respect to stress vs. stress.
Transition strain to full metadynamic
recrystallization
As Figure 5 shows 13), whenever strain during hot
deformation reaches a critical value and then
deformation stops, the static recrystallization
phenomenon that occurs after deformation is called
metadynamic recrystallization. This is because
nucleation happens during deformation. After the
peak, the rate of softening increases continuously and
gets to its maximum value at a particular strain. If
deformation continues up to this specified strain and
then stops, full metadynamic recrystallization occurs.
This is due to the achievement of the maximum rate
of softening which guaranties that nucleus has
formed throughout the specimen. Thus, this certain
strain is called transition strain to full metadynamic
recrystallization state. The value of this strain is
reported 1.5εP for Nb-microalloyed steel 13,14).
Therefore, to estimate this transition strain, the
maximum value of softening i.e. maximum slope of
stress-strain curve at the zone with negative slope
and softening mechanism must be determined. So the
first derivative of
ε
σ
θ
= with respect to σ must be
zero.
)7(0=
⎟
⎠
⎞
⎜
⎝
⎛
σ
ε
σ
d
d
d
d
Fig. 5. Effects of strain on softening mechanisms
during and after deformation:
Zone I-SRX, Zone II-SRX and MDRX, Zone III-
Dynamic and MDRX [13].
by solving Eq. 7 we get the transition strain to
full metadynamic recrystallization as a function of
strain at the maximum stress:
)8(
1
C
P
PT
ε
εε
+=
where C1 is the constant of Eq. 2 that is
determined by Eq. 3. Now, in order to check Eq. 8,
we determine the transition strain for Figure 5 i.e.
for a Nb-microalloyed steel 13). From this curve we
found the approximate values of σS, σP and εP to be
113 MPa, 130 MPa and 0.85, respectively. Also by
assuming K=0.7, the approximate values of σK and εK
are 125 MPa and 0.6, respectively. Inserting these
values into Eq. 3, C1 is found to be 7.013. Eq. 8 gives
the transition strain of 1.2. Thus the ratio of the
transition strain to strain at the maximum stress
equals 1.41 ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛=41.1
P
T
ε
ε
which is comparable to the
approximate value of 1.5 ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛≈5.1
P
T
ε
ε
reported by
other researches for Nb-microalloyed steel 13,14).
Conclusions
A mathematical model was presented to directly
determine the critical strain for onset of DRX and the
transition strain from static-metadynamic
recrystallization to full metadynamic recrystallization
International Journal of ISSI, Vol.4 (2007), No. 1,2
27
after deformation. Numerical results of these
equations are:
1-Critical strain, εC, for onset of DRX by Eq. 6
was found to be 0.46εP to 0.49εP which is in a good
agreement with 0.55εP, reported by other researches
for Nb-microalloyed steel.
2-Transition strain, εT, from static-metadynamic
recrystallization to full metadynamic recrystallization
after deformation is given by Eq. 8 which was
1.41εP for Nb-microalloyed steel that is comparable
to 1.5εP, reported by other investigators.
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Full-text available
  • May 2023
High speed impact at strain rates > 103 s−1 has generic characteristics in context of energy absorption and adiabatic tempera-ture rise. An ultra-high strength low alloy steel (USLA, 3.8mm thick) subjected to high-speed impact has been investigated by focusing on energy absorption driven by thermo-kinetics phenomenon in correlation with the target plate thickness. Result was bulge and reduction in thickness up to 2mm at impact point. An analytical model has been proposed by establishing a relation between absorbed energy and plate thickness to explain deformation characteristics. EBSD analysis was carried out at three specified points, i.e., impact zone (IZ), impact affected zones (IAZ; through-thickness direction near backside) and (IAZ; radial/circumferential direction) with in 3mm diameter. Impact analysis was carried out in Ls Dyna computer code. EBSD and FE (finite element) analysis revealed that the shear wave velocity under the influence of adiabatic temperature rise up to 520K has generated adiabatic shear bands in IZ and IAZ (through-thickness direction), however in the IAZ (radial direction), the plate experienced no adiabatic temperature and increased stress due to compressive wave velocity resulted in greater proportion of recrystallized grain structure which is 25.79% as compared to 10.98% in IAZ(through-thickness direction) and 6.19% in IZ (impact zone). Micro hardness testing of these three distinct zones also revealed that IAZ (radial direction) has higher hardness (630HV) than IZ (600HV) and IAZ (through-thickness direction) (570HV) owing to increased and evenly distributed fine-grained microstructure in IAZ (radial direction), and relatively less deformation damage.
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... However, work hardening rate analysis in conjunction with microstructural studies should be used to reveal the occurrence or absence of DRX. The inflection points in the plots of the work hardening rate (θ = dσ/dε) versus σ can be used to obtain the critical stress for initiation of DRX (σ C ) [163][164][165]. These inflection points are equivalent to a minimum on the plot of -dθ/dσ versus σ [48,49]. ...
A review of hot deformation behavior and constitutive models to predict flow stress of high-entropy alloys
Article
  • May 2022
  • J ALLOY COMPD
This review article summarizes the hot deformation behavior of high entropy alloys (HEAs) and the corresponding constitutive description of flow stress. The potential of hot working for grain refinement via dynamic recrystallization (DRX), reduction of casting defects, and enhancement of mechanical properties of HEAs is explained. The necklace formation, work hardening analysis for identification of the occurrence and initiation of DRX, and the effects of processing parameters on dynamically recrystallized grain size are discussed. The effects of deformation conditions (represented by the Zener-Hollomon parameter), alloying elements, dynamic precipitation, and the presence of phases on the hot deformation behavior and restoration processes of DRX and dynamic recovery (DRV) are overviewed. The application of processing maps for the characterization of the onset of flow instability, cracking, flow softening, and DRX during hot forming of HEAs is presented. Regarding the constitutive modeling of flow stress for characterization of material flow (at different deformation temperatures, strain rates, and strain), the utilization of the threshold stress (due to the presence of phases or their precipitation during high-temperature deformation), and temperature-dependent Young’s modulus, as well as correlating the obtained values of deformation activation energy and stress exponent with the expected ones from the creep theories are taken into account. Afterward, the available methods and equations for modeling and prediction of flow curves during thermomechanical processing are assessed, where the strain-compensated Arrhenius model, artificial neural network (ANN) model, Zerilli-Armstrong model, Johnson-Cook model, Hensel-Spittel model, and dislocation density-based multiscale constitutive model are presented. Finally, some suggestions for future research works are proposed.
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Grain refinement of magnesium alloys by dynamic recrystallization (DRX): A review
Article
Full-text available
  • Aug 2023
For elevated-temperature thermomechanical processing, the occurrence of recrystallization during straining is known as dynamic recrystallization (DRX), which usually happens in Mg alloys during practical hot deformation processes such as rolling, forging, extrusion, friction stir processing (FSP), and multidirectional forging (MDF). Accordingly, the present review paper is dedicated to the grain refinement of Mg alloys by DRX during hot working and summarization of the state-of-the-art. Firstly, the grain refinement of Mg alloys is overviewed, which includes grain refinement achieved during (I) solidification (heterogeneous nucleant particles, growth restriction factor, and ultrasonic treatment), (II) fusion-based and solid-state metal additive manufacturing, (III) annealing of deformed alloys via static recrystallization (SRX), and (IV) severe plastic deformation (SPD). Afterward, the critical conditions for the initiation of DRX, single-peak and multiple-peak (cyclic) flow behaviors, microstructural evolution, and necklace formation during DRX are summarized for Mg alloys. Moreover, the dependency of the DRX grain size and its kinetics to the (I) deformation conditions (temperature and strain rate, as represented by the Zener-Hollomon parameter), (II) initial microstructure (by consideration of continuous and discontinuous DRX mechanisms), (III) alloying elements (solute drag effect), (IV) dynamic precipitation (Zener pinning effect for retardation of grain coarsening), and (V) particle stimulated nucleation (PSN, especially for the Mg alloys containing long period stacking ordered (LPSO) structures and metal-matrix composites) are critically discussed. Furthermore, the metadynamic recrystallization (MDRX) as well as improvement of mechanical properties (represented by the Hall-Petch relationship), enhancement of strength–ductility synergy, and inducing superplasticity for superplastic forming by DRX are overviewed. Finally, the research gaps and distinct suggestions for future works are introduced.
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Kinetics of Metadynamic Recrystallization in Microalloyed Hypereutectoid Steels
Article
Full-text available
  • Jan 2004
The isothermal kinetics of the recrystallization processes of vanadium microalloyed high carbon steels has been measured and modelled. The overall softening data were obtained by double hit compression tests performed at temperatures between 900 to 1 050°C, strain rates of 0.01 to 1 s -1, and inter-pass times of 0.1 to 30s. The recrystallization behavior above and below and the critical strain for dynamic recrystallization was investigated, The results show that there is a transition strain region between where both static and metadynamic recrystallization take place during the inter-pass time. The results also revealed that V and Si have a strong solute drag effect, on the kinetics of metadynamic recrystallization. A kinetic model is proposed which takes the V and Si concentrations into account.
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Initiation of Dynamic Recrystallization in Constant Strain Rate Hot Deformation
Article
  • Jan 2003
In constant strain rate tests, the occurrence of dynamic recrystallization (DRX) is traditionally identified from the presence of stress peaks in flow curves. However, not all materials display well-defined peaks when tested under these conditions. Using plain carbon, Nb-bearing and 321 austenitic stainless steels, it is shown that the onset of DRX can also be detected from inflections in plots of the strain hardening rate 0 against stress a or, equivalently, from inflections in In theta-ln sigma and ln sigma-epsilon plots regardless the presence of stress peaks in the flow curves. These observations are verified by means of metallography. A unified description of the flow curve is introduced based on normalization of the stress and strain by the respective peak or steady state values. This approach reveals that, in a given material, the ratio of DRX critical stress to the peak or steady state stress is constant, as is that of the critical strain to the corresponding strain values. Furthermore, it is shown that the present technique can be used to establish the occurrence of DRX when this cannot be determined unambiguously from the shape of the flow curve.
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Comparison between Static and Metadynamic Recrystallization. An Application to the Hot Rolling of Steels
Article
  • Jan 1997
An investigation has been performed to simulate the microstructural evolution during the hot strip rolling of steels. During this research, the controlling softening,and recrystallization mechanisms after the hot deformation of austenite were first determined using single-hit,and double-hit compression techniques. Based on the experimental flow and softening data, several mathematical expressions have been proposed to quantify the boundary conditions and overall kinetics for static and metadynamic recrystallization, respectively. The static model is expressed as a function of initial grain size, strain, strain rate and temperature, while the metadynamic one only depends on the strain rate and temperature. Together with industrial mill processing parameters, these models were incorporated into an integrative analysis of the hot rolling of plain carbon steel strips. The simulation results indicate that metadynamic recrystallization is dominant and leads to the full softening during rough rolling, where processing temperatures are high and strain rates relatively low. Metadynamic recrystallization can also occur between the initial finishing stands, when larger reductions are applied to the steel band. in general, however, static recrystallization becomes more and more important in the finishing mill. Partial static recrystallization may take place in the later stages of finish rolling, which can be attributed?ed to the decreasing processing temperatures, reduced stand strains and much shorter interstand times. The evolution of the austenite microstructure during hot rolling can be characterized by the grain refinement associated with recrystallization and the subsequent grain growth. Although grain growth is significant during rough rolling, grain refinement with minimum interstand grain growth plays a key role during finish rolling.
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Constitutive equations for modelling flow softening due to dynamic recovery and heat generation during plastic deformation
Article
  • Feb 1998
  • MECH MATER
The flow softening of metals at elevated temperatures is modelled in this paper based on the theory of viscoplasticity. A single internal state variable constitutive model is presented to represent the deformation behavior of metals that exhibit flow softening caused by the competing hardening and recovery processes and heat generation during plastic deformation. The constitutive equations are validated by hot torsion testing that has advantage over tension and compression in generating flow stress to large strains. The material constants determination methods have been developed and successfully applied to aluminium alloy 5083 over a wide range of strain rates and temperatures. The prediction of the flow stress in aluminium alloy 5083 as a function of strain rate, temperature and strain has shown to be in good agreement with the test data.
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An investigation on the effect of carbon and silicon on flow behavior of steel
Article
  • May 2002
  • MATER DESIGN
The effect of carbon and silicon on the occurrence of dynamic recrystallization as well as on the flow stress of steel was investigated. For this purpose, three grades of steels including low carbon, medium carbon and high silicon low carbon steel were examined. Single hit experiments at temperatures of 900–1200 °C and strain rate of 0.01–2 s−1 were performed to determine the phase transformation kinetics. Different socking times were utilized to detect the effect of initial austenite grain size on the kinetics of dynamic recrystallization and flow behavior of the steels. It was found that increasing the carbon content leads to a higher rate of dynamic recrystallization at high temperatures and low strain rates, while silicon causes postponement of the occurrence of dynamic recrystallization. Moreover, increasing the initial austenite grain size postpones the onset of dynamic recrystallization and leads to a higher flow stress at equivalent deformation conditions.
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