Modeling high temperature flow stress of
pure copper using a physically based
model for dynamic recrystallization
Ho Won Lee1, *, Young-Seon Lee1, Seong-Hoon Kang1, Yong-Taek Im2
1
Korea Institute of Materials Science, 797 Changwondaero, Seongsan-gu, Changwon, Gyeongnam 642-831, Republic of Korea,
2
Computer Aided Materials Processing Laboratory, Department of Mechanical Engineering, KAIST, 291 Daehak-ro,
Yuseong-gu, Daejeon 305-701, Republic of Korea
In the present study, a new high temperature flow stress model considering dynamic recrystallization phenomena was developed using a physically based model. In the model,
the recrystallized microstructure was assumed to be a set of grain aggregates. For each
grain aggregate, changes of dislocation density and grain size were calculated using three
sub-models for work hardening, nucleation, and nucleus growth. In the case of modeling
for the work hardening process, a conventional single parameter dislocation density model
was used to calculate change of dislocation density in grains. For modeling nucleation, constant nucleation rate model and nucleation criterion developed by Roberts and Ahlblom
was used. It was assumed that the nucleation occurs when the dislocation density of certain grain reaches a critical nucleation criterion. Conventional rate theory was used to
model nucleus growth. The predicted results are compared with experimental findings and
simulated results by cellular automata on dynamic recrystallization of pure copper. The
general characteristics of dynamic recrystallization such as single to multiple peak conversion in flow stress and power-law relationship between stress and grain size was reproduced and compared with other results quantitatively. This study would facilitate a comprehensive understanding of the dynamic recrystallization process during hot deformation.
Keywords: dynamic recrystallization, flow stress, hot deformation, physically based model
1. INTRODUCTION 1
Dynamic recrystallization (DRX) is one of the key processes that governs the final microstructure during hot deformation. Metals with low or medium levels of stacking fault
energy such as copper and nickel undergoes the DRX while
metals with high stacking fault energy undergoes dynamic
recovery. During the DRX process, new grains originate at
high-angle grain boundaries such as prior grain boundaries,
the boundaries of recrystallized grain, and deformation
bands and twins. The new grains grow to the high disloca-
* Author to whom correspondence should be addressed.
tion density of the original grain due to the driving force,
which is driven by the difference in the dislocation density.
However, recrystallized grains will cease to grow as the
material deforms further, resulting in reducing the dislocation density difference. The resulting microstructure and
material behavior during the DRX vary with the deformation condition. Differences in grain size, texture, material
behavior affect in final mechanical properties and deformation aspect of final product. Therefore, predicting the
DRX better is essential when attempting to control the mechanical properties and deformation process of final product.
Various models have been proposed thus far to predict microstructural changes and material behavior during the DRX.
Typically, the Johnson-Mehl-Avrami-Kolmogrov (JMAK)
theory [1] was widely used to model homogeneous recrystallization kinetics. The empirical mathematical model was
widely used to model the flow behavior and microstructure
for industrial applications [2-4]. In these works, microstructure evolution was described as a function of strain, strain
rate, temperature, and initial grain size. However, these applications only concentrated on the hot strip and plate rolling
processes because of the difficulties that arise in exact thermo-mechanical modeling of bar rolling or hot forging processes. Therefore, such modeling was extended by integrating with a finite element analysis to provide more realistic
description of local process parameters such as strain, strain
rate, and temperature [5-7]. However, these methods predicted local microstructure as an average grain size only and
local heterogeneity was not considered properly. In addition,
it is not applicable to complex problem because of phenomenological characteristics.
To describe heterogeneous nature of the DRX and predict
local microstructure evolution in detail, many mesoscale
modeling methods such as the Monte Carlo model [8, 9], the
phase field model [10, 11], and the cellular automata model
[12-14] have been proposed as well. Although these models
successfully described microstructure evolution during the
DRX, most studies have focused on the DRX of ideal process like isothermal compression with a constant strain rate
because of difficulty in modeling the deformation.
Recently, there have been attempts to combine mesoscale
microstructure simulation with macroscale finite element
method (FEM) for describing deformation during DRX
more accurately. Lee et al. [15] combined cellular automata
and the finite element method to predict microstructural
changes by DRX during a nonisothermal hot forging process
and validated the proposed model by comparing its results
with the experimental results obtained by electron backscattered diffraction (EBSD). The proposed model generally
predicted microstructural changes accurately. However,
industrial application of this method is difficult owing to the
long calculation time due to the large scale.
To overcome the limitation of JAMK model and multiscale
model, grain scale approach using physically based model
has been proposed recently [16, 17]. In these models, the
dynamic recrystallization was modeled by using grain as a
representative volume element and mean field theory.
Therefore, in this study, we aim to develop a new physically
based constitutive model that can predict microstructural
changes and to use this model for industrial applications by
combining it with FEM. In the current model, the grain aggregate will be used as a representative volume element
different with previous model. This model will provide a
reduced calculation time as compared to that required for
simultaneous analysis by cellular automata and the finite
element method.
Fig. 1 Schematic illustration of dynamically recrystallized microstructure
Fig. 2 Flow chart of the current model for calculating dynamic
recrystallization process
2. DYNAMIC RECRYSTALLIZATION MODEL
2.1 Model Structure
In the present study, the dynamically recrystallized microstructure of polycrystalline metal was represented by a set of
grain aggregates. The local microstructure during DRX is
totally inhomogeneous. The microstructure has both heterogeneously deformed grains depending on its orientation and
recrystallized grains with different cycles of DRX as shown
in Fig. 1. Therefore, in the current study, the microstructure
will be represented by certain number of representative
grain aggregate with different Taylor factor and different
cycle. Each grain aggregate has state variables of a specific
orientation or the Taylor factor, dislocation densities ( ρ[ j ,i ] ),
number of grain in the same aggregate ( N[ j .i ] ), surface
boundary area ( S[ j ,i ] ), sum of grain diameter in the grain
aggregate ( D[' j ,i ] ) and volume fraction ( f[ j .i ] ).
The model numerically calculated DRX with the following
steps as shown in Fig. 2: Calculation of the minimum time
step to ensure better results; calculation of dislocation density changes by work hardening at each time step; nucleation
of the recrystallization embryo by comparing the dislocation
densities in the grain boundaries with the critical dislocation
density; simulation of the growth of the nucleus to the high
dislocation density area. The details of each process will be
given in the following sections. This approach enables reduced calculation time compared with conventional method
which used every grain as a representative volume element.
2.2 Dislocation Density Evolution
To model the dislocation density changes by work hardening, the ‘one-parameter’ model by Kocks and Mecking [18]
was used in the current study due to its simplicity and applicability in a hot deformation process. The KocksMecking (KM) model is based on the assumption that the
average dislocation density determines the kinetics of the
plastic flow. In the kinetic equation for hot deformation, the
flow stress is proportional to the square root of the dislocation density, as follows:
σ [ j ,i ] = αµ b ρ[ j ,i ]
(1)
In this equation, α is a numerical constant, µ denotes the
Fig. 3 Schematic illustration of nucleation process
2.3 Nucleation
Nucleation can happen when the stored energy due to dislocation accumulation can overcome the boundary energy
related to the curvature. In the present investigation, the
critical dislocation density proposed by Roberts and
Ahlblom [19] was used.
 20γε 
ρc = 
2 
 3blM τ 
1/3
In this equation, l is the dislocation mean free path, M
represents the grain boundary mobility, τ is the dislocation
line energy (expressed as τ = 0.5gb 2 ), and γ is the grain
boundary energy.
In Eq. (3), the grain boundary mobility is given, as shown
below [20].
shear modulus, b is the magnitude of Burger’s vector of
dislocation, and ρ[ j ,i ] is the average dislocation density of
grain aggregate [ j , i ] . Where, i represents DRX cycle and
j represent Taylor factor number.
The change in the dislocation density of a fine-grained material may be considered to consist of two components, as
follows:
d ρ[ j ,i ]
= k1 − k2 ρ[ j ,i ]
dε
(2)
(3)
M
=
δ D0b b
kT
 Q 
exp  − b 
 RT 
(4)
Here, δ is the characteristic grain boundary thickness, D0b
denotes the boundary self-diffusion coefficient, Qb is the
activation energy for boundary diffusion, k represents the
Boltzmann’s constant, and R is the gas constant.
The nucleation rate was assumed to be constant during the
whole DRX process and a function of temperature. Then,
number of nuclei for grain aggregate [ j , i ] can be defined as
follows:
Here, k1 = 1 / bl and k2 is a function of temperature and
strain rate, k2 = f (T , ε ) . The k1 and k2 values were acquired
from the isothermal hot compression results available in the
literature [14].
Q
N[ j ,i ] nε m exp  nucl
=
 RT

 ⋅ f[ j ,i ] ⋅ ∆t

(5)
where n , m , Qnucl are constants and ∆t is a time increment.
As shown in Fig. 3, new grain of grain aggregate [ j , i + 1] is
nucleated at the prior grain of grain aggregate [ j , i ] .
So, the changes of volume fraction in the grain aggregate
[ j , i ] during nucleation can be described as multiplication of
number of nuclei and volume fraction of nuclei compared to
the total volume as follows:
∆f[ j ,i ] =
− N[ j , i ] ⋅
V0
Vtot
(6)
where V0 is the volume of nucleus and Vtot .is the total volume.
The sum of grain diameter and surface boundary area
changes compared to the volume fraction during nucleation
while dislocation density of deformed aggregate did not
change as follows.
′
′
∆D=
[ j ,i ] D[ j ,i ] ⋅ ∆f[ j ,i ]
∆S[ j ,i ]
∆f[ j ,i ]
=
−
⋅ S[ j ,i ]
f[ j , i ]
t
t
ρ[t +∆
j ,i ] = ρ[ j ,i ]
(7)
(8)
(9)
The dislocation density of newly formed nucleus ( ρ0 ) is
almost zero although it will change during further deformation. Therefore, the dislocation density at the time t + ∆t
can be described as follows.
t
ρ[t +∆
j , i +1] =
ρ[t j ,i +1] ⋅ f[ tj ,i +1] + ρ0 ⋅ ∆f[ tj ,i +1]
f[ tj ,i +1] + ∆f[ tj ,i +1]
Fig. 4 Schematic illustration of growth process during dynamic
recrystallization
to the pressure, the constant of proportionality being the
mobility (M) of the boundary as follows [1] :
=
v M ⋅P
(13)
P = τ ( ρ[l .k ] − ρ[ j .i ] ) −
2γ
r
(14)
However, the second term in Eq. (14) is omitted in the current model because it is negligible compare to the first term.
During nucleus growth, the volume fraction of grain aggregate changes as follows by considering volume ( S[l , k ] v∆t )
swept by nucleus surface boundary:
∆f[ j ,i ] = −∆f[l , k ] =
S[ j ,i ] v∆t
Vtot
f[ l , k ]
(15)
(10)
In this equation, the changes of volume fraction between
aggregate [ j , i ] and [l , k ] are only considered by multiply-
The sum of grain diameter and surface boundary area was
calculated using following equation by assuming the spherical grain shape.
∆D[′j ,i +1] =N[ j ,i ] ⋅ D0
(11)
N π D0 2 ⋅ Vtot
∆S[ j ,i +1] =⋅
(12)
where Do represents the diameter of nucleus.
2.4 Nucleus Growth
The driving force for nucleus growth of dynamically recrystallized grains is the difference of stored strain energy between recrystallized grains and prior deformed grains as
shown in Fig. 4. A grain boundary moves with a velocity ( v )
in response to the net pressure ( P ) on the boundary. It is
generally assumed that the velocity is directly proportional
ing volume fraction of aggregate [l , k ] to avoid duplication
of calculation.
The sum of grain diameter in the aggregate was calculated
using following equations.
∆D[′j ,i=
] 2v ⋅ ∆t ⋅ n[ j ,i ] ⋅ f[l , k ]
(16)
D[′l , k ] ⋅ S[ j ,i ] ⋅ v ⋅ ∆t
∆D[′l , k ] =
f[ l , k ]
Vtot
(17)
The change of surface boundary area during nucleus growth
was calculated as follows:
∆S[ j ,=
4π ⋅ D[′j ,i ] f[l ,k ]v ⋅ ∆t
i]
(18)
∆f[ j ,i ] ⋅ S[l ,k ]
∆S[l ,k ] =
−
f[ l , k ]
(19)
Fig. 5 The comparison of flow stress curves between the experimental and physically based model
Fig. 7 Flow curve (dot) and recrystallized fraction (line) at the
high Zener-Hollomon parameter
Fig. 6 The comparison of average recrystallized grain sizes between the experimental and physically based model
Fig. 8 Flow curve (dot) and recrystallized fraction (line) at the
high Zener-Hollomon parameter
3. RESULTS AND DISCUSSION
3.1 Validation of Developed Model
To verify the developed physically based model, flow stress
curves and final grain sizes at various temperatures and
strain rates were compared with the experimental results in
the literature [14], as shown in Fig. 5 and Fig. 6, respectively. The proposed model predicted the flow stress generally
well, although it exaggerated the oscillating flow curves at
600 oC and 0.01 s-1.
This occurred because the characteristics of dynamic recrystallization in the single- and multi-peak curves were rather
different from each other. To represent both curves accurately, the model should be considered differently in some manner. The predicted grain sizes were generally in good
agreement with the experimental results, although some
error existed in the results.
Fig. 9 Changes of average grain size during dynamic recrystallization for the single- and multi-peak flow curves
3.2 Characteristics of Flow Curves
The shape of the flow stress curves is related to the ratio of
the prior and recrystallized grain size. The flow stress curve
at high Zener-Hollomon parameter (Z) showed a broad sin-
gle peak (e.g., 500 oC and 0.1 s-1 in Fig. 5). On the other
hand, the flow stress curves at low Z values showed oscillating multi-peak curves under low strain and steady curves
under high strain (e.g., 700 oC and 0.1 s-1 in Fig. 5). It is
known that single- and multi-peak curves are related to the
prior and final grain sizes. It is known that if the prior grain
size is greater than two times the recrystallized grain size, a
single-peak curve may occur [21]. Therefore, the characteristics of the flow curves were investigated in detail in this
study.
The single-peak flow curve and recrystallized fraction during DRX for the single-peak and multi-peak curves are represented in Fig. 7 and Fig. 8, respectively. In Fig. 7, the
slope of the flow stress was changed past the critical strain
(about 0.47) and reached its peak at a strain of about 0.5.
The flow curves then softened and hardened again from the
strain value of 0.65 and recrystallized fraction of 0.7. This
re-hardening is related to low nucleation rate due to the
small grain boundary area at the higher recrystallized fraction. The average grain size curve changed rapidly due to
the DRX process as shown in Fig. 9. The average grain size
was held constant up to the critical strain, decreased rapidly
directly past the critical strain, and then became saturated.
The rapid decrease in the average grain size is due to the
difference of recrystallized grain size and initial grain size.
In Fig. 8, the flow stress also showed a curve similar to that
of the single-peak DRX up to a strain of 0.1. However, the
flow stress increased again past a strain value of 0.12 owing
to the lack of nucleation between the DRX cycles. The next
DRX cycle was then initiated past a certain deformation,
and the stress was decreased again by further nucleation.
The average grain size was also oscillating owing to the
several cycles of the DRX that were run as shown in Fig. 9.
The average grain size initially decreased immediately after
the initiation of the current cycle of the DRX as a consequence of the nucleation stage. It then increased rapidly due
to the growth and decreased again by the next DRX cycle.
4. CONCLUSION
In the present study, the physically based dynamic recrystallization model was successfully developed using grain aggregate as a representative volume element. It predicted
general tendency of dynamic recrystallization in flow stress
and average grain size although it showed some errors in
specific conditions. The single-peak and multi-peaks flow
curves due to grain refinement and coarsening phenomena
were also investigated. In terms of grain refinement, the
flow stress curves showed both a single peak, although a
multi-peak curve was noted during the grain coarsening
phenomena. This grain aggregate model will be more useful
to combine with finite element analysis for simulating industrial process compared to the conventional grain scale
approach because of its fast calculation speed and less
memory use.
Acknowledgments: This work was supported by the Basic
Research Program of the Korea Institute of Materials Science. One of the authors (H.W. Lee) also appreciated financial support from the second stage of the Brain Korea 21
Project in 2011 at the KAIST Mechanical Engineering Research Institute.
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