Constitutive model for high temperature

Mechanics of Materials 42 (2010) 157–165

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Mechanics of Materials
journal homepage: www.elsevier.com/locate/mechmat

Constitutive model for high temperature deformation of titanium
alloys using internal state variables
Jiao Luo, Miaoquan Li *, Xiaoli Li, Yanpei Shi
School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710072, PR China

a r t i c l e

i n f o

Article history:
Received 14 October 2008
Received in revised form 20 July 2009

Keywords:

Titanium alloy
Constitutive model
Genetic algorithm
Flow stress
Grain size
Dislocation density

a b s t r a c t
The internal state variable approach nowadays is more and more used to describe the
deformation behavior in all of the metallic materials. In this paper, firstly the dislocation
density rate and the grain growth rate varying with the processing parameters (deformation temperature, strain rate and strain) are established using the dislocation density rate
as an internal state variable. Secondly the flow stress model in high temperature deformation process is analyzed for each phase of titanium alloys, in which the flow stress contains
a thermal stress and an athermal stress. A Kock–Mecking model is adopted to describe the
thermally activated stress, and an athermal stress model is established using two-parameter internal state variables. Finally, a constitutive model coupling the grain size, volume
fraction and dislocation density is developed based on the microstructure and crystal plasticity models. And, the material constants in present model may be identified by a genetic
algorithm (GA)-based objective optimization technique. Applying the constitutive model to
the isothermal compression of Ti–6Al–4V titanium alloy in the deformation temperature
ranging from 1093 to1303 K and the strain rate ranging from 0.001 to 10.0 s1, the 20
material constants in those models are identified with the help of experimental flow stress
and grain size of prior a phase in the isothermal compression of Ti–6Al–4V titanium alloy.

The relative difference between the predicted and experimental flow stress is 6.13%, and
those of the sampled and the non-sampled grain size are 6.19% and 7.94%, respectively.
It can be concluded that the constitutive model with high prediction precision can be used
to describe the high temperature deformation behavior of titanium alloys.
Ó 2010 Published by Elsevier Ltd.

1. Introduction
Titanium alloys are used for a wide variety of aerospace
applications owing to their unique combination of
mechanical and physical properties, i.e., high specific stiffness and strength at ambient and elevated temperatures,
excellent corrosion and oxidation resistance, and good
creep resistance (Sen et al., 2007). However, these alloys
are more difficult to fabricate than other metallic materials
due to the high flow stress at elevated temperatures and
are strongly sensitive to the processing parameters such
as strain, strain rate and deformation temperature, and/or
* Corresponding author. Tel.: +86 29 88491478.
E-mail address: honeymli@nwpu.edu.cn (M. Li).
0167-6636/$ - see front matter Ó 2010 Published by Elsevier Ltd.
doi:10.1016/j.mechmat.2009.10.004

material composition in the forming process. Therefore,
reliable constitutive equation, which describes the correlation of dynamic material performance with processing
parameters, needs to be developed in order to understand
the deformation behavior in depth and optimize the deformation processes.
In the past several decades, a number of efforts had
been focused on the constitutive modeling and analysis
of plastic flow for metals and alloys in high temperature
deformation (Zener and Hollomon, 1944; Kocks and Maddin, 1956; Sellars and McTegart, 1966; Shida, 1969; Vinh
et al., 1979; Johnson and Cook, 1983, 1985; Klopp et al.,
1985). The empirical and semi-empirical regression models were mostly obtained. But, those regression models
are unsatisfied because the complicated and non-linear

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J. Luo et al. / Mechanics of Materials 42 (2010) 157–165

relationships may exist between flow stress, microstructure and processing parameters in the high temperature
deformation. The artificial neural network (ANN) method
unlike the regression method, however, does not need a

mathematical formulation and has the capability of selforganization or ‘‘learning”. This approach is especially suitable for treating non-linear phenomena and complex relationships and has been successfully applied to the
prediction of constitutive relationships of a few alloys (Li
et al., 1998, 2006; Cetinel et al., 2002; Kumar et al.,
2007). However, the successful application of ANN model
is strongly dependent on the availability of extensive, the
high quality data and characteristic variables, and the
modeling offers no physical insight. Therefore, the further
application to establish the constitutive equations is
limited.
With further understanding of plastic deformation
mechanism, physically based constitutive equation has
been developed. Zerilli and Armstrong (1987) proposed
two constitutive equations for body centered cubic (bcc)
and face centered cubic (fcc) materials, which incorporated
information regarding the thermally activated motion of
dislocations. Follansbee and Gray (1989) proposed a mechanism-based model which focused on the low temperature
material behavior and considered the deformation to be
controlled by thermally activated process only. For a review of more recent development in these models refer
to Donahue et al. (2000), Picu and Majorell (2002), Kim
et al. (2003, 2005) and Voyiadjis and Almasri (2008). In recent years, a number of internal state variable (ISV)-based

constitutive models have been proposed in the literature
for the elevated temperature, rate-dependent deformation
of metals (Brown et al., 1989). Zhou and Clode (1998) presented a single internal state variable constitutive model to
represent the deformation behavior of metals that exhibited flow softening caused by the competing hardening
and recovery processes and heat generation during plastic
deformation. Roters et al. (2000) introduced a new workhardening model for homogeneous and heterogeneous
cell-forming alloys based on three internal state variables.
Garmestani et al. (2001) presented a transient model based
on Hart’s original model. The new model introduced a new
state variable in the form of a micro-hardness parameter.
Lee and Chen (2001) formulated the constitutive relationships of thermo-visco-elastic–plastic continuum theory in
Lagrangian form, and three internal state variables (plastic
strain tensor, back strain tensor and a scalar hardening
parameter) were also incorporated. Lin et al. (2005)
developed a set of mechanism-based unified viscoplastic
constitutive equations which modeled the evolution of dislocation density, recrystallization and grain size during and
after hot plastic deformation. The physical based internal
state variable approach instills greater confidence than
the empirical method, and provides the greatest potential
for enhancing scientific understanding (Grong and Shercliff, 2002).

For the internal state variable-based constitutive equation, it is critical to select state variables. Quantities such as
stored energy and flow stress are not state variables due to
their direct dependence on the underlying microstructure.
It is well known that the microstructure evolution affects

strongly the deformation behavior and mechanical properties of material. Therefore, an appropriate representation
of mechanical behavior has to be based on the microstructural state variables that are affected by the process history
of material (Li and Li, 2006). Recently, a number of
researchers (Li et al., 1995; Busso, 1998; Estrin, 1998; Estrin et al., 1998; Fedelich, 1999; Goerdeler and Gottstein,
2001; Karhausen and Roters, 2002; Ganapathysubramanian and Zabaras, 2004; Lin and Dean, 2005; Beyerlein and
Tomé, 2008) chose the microstructural state variables,
such the dislocation density and grain size as internal state
variables in the constitutive equations. These works are
very helpful for understanding the physical mechanisms
during the plastic deformation and designing the processing parameters. However, there is no constitutive model
available to model interactive relationships between grain
size, volume fraction, dislocation density and deformation
behavior of material. Therefore, a multi-scale model will be
established in present research.
In this paper, firstly a microstructure model including

dislocation density rate and grain growth rate is established using the dislocation density rate as an internal state
variable. Secondly, the mechanisms and the driving forces
for the deformation behavior and the effect of microstructure evolution on the viscoplasticity of material are
analyzed. And, a constitutive equation, in which the dislocation density and grain size of matrix phase are taken as
internal state variables, is developed based on the physically microstructure model. Applying these models to the
isothermal compression process of Ti–6Al–4V titanium
alloy, the material constants in these models are determined by a genetic algorithm (GA)-based objective optimization technique.
2. Microstructure model
2.1. Dislocation density rate
It is well known that dislocation density q in high temperature deformation of metals and alloys depends on two
competing processes: working hardening and dynamic
softening. Mecking and Kocks (1981) had pursued a phenomenological approach (the KM model) to predict the
variation of dislocation density with strain for stage III
hardening of metals. The model is based on the assumption
that the kinetics of plastic flow is determined by a single
structural parameter (dislocation density q) which represents the entire current structure (Ding and Guo, 2002).
In the KM model, the dislocation storage rate is proportional to q1/2, and the dislocation annihilation rate is proportional to q, so the variation of dislocation density
with strain can be written as:

pffiffiffiffi

dq
¼ k1 q  k2 q
dep

ð1Þ

where q is the average dislocation density, ep is the plastic
strain, k1 is a material constant and describes the storage of
dislocations at the dislocation forest obstacles, k2 describes
dynamic recovery of the dislocation density, essentially by
annihilation of pairs of dislocation segments with opposite

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J. Luo et al. / Mechanics of Materials 42 (2010) 157–165

Burgers vectors, which is a function of deformation temperature and strain rate, and can be described by (Picu
and Majorell, 2002):

k2 ¼ k20

 Q dm
4 2
e_ 0 
1  e0:7Rc q e RT
_e

ð2Þ

where R is the gas constant (8.314 J/(mol K)), Qdm is the
activation energy for cross-slip and recombination (kJ/
mol), k20 is a proportional constant, e_ 0 and e_ are a reference
and the applied strain rate, and Rc is a cut-off radius beyond which dislocation cannot cross-slip and recombine.
In Eq. (2) a thermally activated term (exp(-Qdm/RT)) is
introduced owing to both cross-slip and climb are thermally activated processes, which reflects temperature
dependence of the evolution of dislocation population at
given plastic strain. At the low temperature, the recovery
term plays a minor role and hence the dislocation density
is insensitive to the deformation temperature. But at high
temperature, the effect of deformation temperature on

the dislocation density is not negligible (Picu and Majorell,
2002).
When the flow stress reaches the steady value, the dis4 2
location density q is so large that e0:7Rc q in Eq. (2) is close
to zero. Therefore, Eq. (2) is simplified to the following
equation:

k2 ¼ k20

e_ 0 QRTdm
e
e_

ð3Þ

Combining Eqs. (1) and (3), the present dislocation density rate q_ in high temperature deformation can be expressed as:

pffiffiffiffi

Q dm

q_ ¼ a1 qje_ j  a2 e RT q

ð4Þ

where a1(k1) and a2ðk20 e_ 0 Þ are material constants. The
right hand side of Eq. (4) has two terms, the first one characterizes the processes of dislocation storage, and the second characterizes the concurrent dislocation annihilation
by dynamic recovery.

where D is the boundary self-diffusion coefficient (m2/s),
which is an exponential function of deformation temperature. d is the characteristic grain boundary thickness (m),
b is the Burgers vector magnitude (2.86  1010 m), D0 is
the self-diffusion constant, k is Boltzmann’s constant
(1.381  1023 J/K), and Qpd is the boundary diffusion activation energy (kJ/mol). In Eq. (6) the effect of the exponential term is larger than that of 1/T, hence the grain
boundary mobility increases with an increase of deformation temperature.
Substituting Eq. (6) into Eq. (5), the static grain growth
rate d_ static can be written as:
Q

pd
c
d_ static ¼ b0 d 0 T 1 e RT

where c0 and b0 (dD0rsurfb/k) are the temperature-dependent material constants. The right hand side of Eq. (7) reflects temperature dependence of the static grain growth.
In high temperature deformation, the dynamic grain
growth induced by plastic strain can be expressed as (Dunne, 1998; Zhou and Dunne, 1996):
c
d_ dyn ¼ b1 je_ jd 1

c
d_ dis ¼ b2 q_ c3 d 2

d_ static

Mrsurf
¼
d

ð5Þ

where d_ static is the static grain growth rate, d is the average
grain size (lm), rsurf is the grain boundary energy per unit
area (J/m2), and M is the grain boundary mobility (m4/(J s)),
which can be written as (Ding and Guo, 2002):

M¼

d_ ¼ d_ static þ d_ dynamic þ d_ dis
Q pd

c1

T 1 e RT þ b1 je_ jd

c2

 b2 q_ c3 d

ð10Þ

If the temperature rise is ignored in isothermal deformation processes, i.e. T is a fixed constant, the grain growth
rate, namely Eq. (10) can be written as:

d_ ¼ d_ static þ d_ dynamic þ d_ dis
c0

¼ b0 d

þ b1 je_ jd

c1

c2

 b2 q_ c3 d

ð11Þ

In summary, a microstructure model of metals and alloys in high temperature deformation, with the dislocation
density rate being an internal state variable, can be expressed as:

pffiffiffiffi

Q dm

q_ ¼ a1 qje_ p j  a2 e RT q
Q

pd
c
c
c
d_ ¼ b0 d 0 T 1 e RT þ b1 je_ jd 1  b2 q_ c3 d 2

ð12Þ

3. Physical-based constitutive model
3.1. Plastic deformation mechanisms

Q pd



dDb dD0 e
¼
kT
kT

ð9Þ

where b2, c2 and c3 are material constants. The dislocation
density is treated as an internal state variable.
Considering the static grain growth, the dynamic grain
growth induced by plastic strain, and the effect of the dislocation density on grain growth, the grain growth rate
may be taken the form of

c0

In high temperature deformation of titanium alloys, the
grain growth rate of the prior a phase is composed of the
static grain growth, dynamic grain growth induced by plastic strain, and grain growth due to the variation of dislocation density (Luo et al., 2008).
The static grain growth as an atomic diffuse process affected by thermal effect relates to grain boundary mobility.
Therefore, the static grain growth is given by the following
expression (Shewmon, 1969):

ð8Þ

where b1 and c1 are the temperature-dependent material
constants.
The effect of dislocation density on grain size is considered in high temperature deformation of titanium alloys,
therefore, the average grain size can be described by the
following equation (Li and Li, 2005):

¼ b0 d

2.2. Grain growth rate

ð7Þ

RT

b

ð6Þ

Multiple deformation mechanisms are active in high
temperature deformation of titanium alloys. Stress is com-

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puted using Taylor’s assumption by which the different
mechanisms act in parallel, contributing to stress separately. The total stress is assumed to be composed of a
thermally activated stress s* and an athermal stress sl
(Picu and Majorell, 2002).

s ¼ s þ sl

ð13Þ

where, essentially s* is due to the short-range thermally
activated effect which may includes the Peierls stress,
point defects such as vacancies and self-interstitials, other
dislocations which intersect the slip plane, alloying elements, and solute atoms (interstitial and substitutional).
The athermal stress sl is mainly due to the long-range effects such as the stress field of dislocation forests and grain
boundaries (Nemat-Nasser et al., 2001).
The thermally activated stress s* is given by (Kocks,
1976; Mecking and Kocks, 1981):


s ¼s

0

"

RT
c_ 0
1
ln
DG
c_


1=q #1=p

;

ð14Þ

where DG is the activation energy for deformation (kJ/
mol), s0 is the mechanical threshold stress (MPa), or the
value of the thermal stress at 0 K, R is the gas constant
(8.314 J/(mol K)), p, q and c_ 0 are material constants. The
reference rate c_ 0 is related to the vibrational frequency of
dislocations arrested at an obstacle, or the obstacle overcoming attempt frequency (Kocks et al., 1975).
While traditionally the thermally activated stress captures the reference rate and temperature dependence of
the flow stress, the athermal stress accounts for strain
hardening (Picu and Majorell, 2002). A two-parameter
internal variable model is developed to represent the
athermal stress. In this paper, the dislocation density q
and the grain size d are treated as internal state variables.
The athermal stress is given by:

pffiffiffiffi

sl ¼ alðTÞb q þ Kd1=2

ð15Þ

where, b is the Burgers vector magnitude (2.86  1010 m),
a and K are material constants, q is the dislocation density
and d is the average grain size (lm). The second term in Eq.
(15) is the standard Hall Petch term which stands for the
strengthening effect due to grain boundaries. The constant
K is taken 12.7 MPa mm1/2 in a-Ti (Courtney, 2000), and
a = 0.5. l(T) is the temperature-dependent shear modulus,
and is expressed as (Varshni, 1970):

lðTÞ ¼ l0 

e
expðT r =TÞ  1

ð16Þ

where e and Tr are material constants.
In the earlier researches about constitutive model of
plastic deformation, the average grain size d is assumed a
constant which is equal to the initial grain size at the
beginning of deformation. In fact the microstructure of
alloys undergoes a series of dynamic changes in high temperature deformation. Therefore, the effect of microstructure evolution on macroscopic deformation behavior in
the present study is adopted and the average grain size d
is treated as an internal state variable in Eq. (15). The grain
size varies with the processing parameters and is determined through Eq. (12).

3.2. Constitutive model for two-phase titanium alloy
For two-phase titanium alloys, the microstructure below the b transus temperature consists of a phase (hexagonal close-packed, hcp) and b phase (body-centered cubic,
bcc). When the grain sizes of a phase and b phase are in the
same order of magnitude, moreover a phase and b phase
both possess the characteristic of plastic flow, the plastic
deformation is dependent on the volume fraction of phase.
Assuming the strain in each phase is equal to the macroscopic applied strain, the overall plastic stress on the
mechanism level (shear stress) is expressed by a rule of
mixture (ROM) as follows (Kim et al., 2001):

s ¼ fa sa þ fb sb

ð17Þ

where fa and fb are the temperature-dependent volume
fractions of a phase and b phase, respectively, and
fa + fb = 1, sa and sb are the stresses of a phase and b phase,
respectively.
Using a standard rule of mixture for the total stress is
consistent with Taylor’s assumption by which the total
stress results by a superposition of strengthening effects
due to various mechanisms. It is however known that such
an approximation is not accurate in the high temperature
regime of interest here (Picu and Majorell, 2002), because
above-mentioned rule of mixture is established on the basis of an ideal assumption of two aligned continuous
phases under iso-strain conditions. Strengthening exceeding the ROM averages has been observed in the deformation of in situ Cu/M composites (Funkenbusch and
Courtney, 1985). This has been attributed in part to the difficulties of the dislocation slip across interfaces between
the different phases or the different grains and the strain
hardening behavior of the individual phases. Thus, the
modified rule of mixture is expressed as:

s ¼ n1 fa sa þ n2 fb sb

ð18Þ

where n1 and n2 are the strengthening coefficients that are
greater than 1. Finally, the variation of the b phase volume
fraction fb(T) with temperature is an internal state variable
that needs to be specified. The (a + b) phase transformation
may be described by an Avrami equation. Then, fa(T) and
fb(T) may be obtained directly from the phase diagram as:

8
 
< f ðTÞ ¼ T w T < T
b
sus
T sus
:f ¼ 1  f
a
b

fb ¼ 1 T  T sus
fa ¼ 0

ð19Þ
ð20Þ

where w is a fitting constant, Tsus is the b transus
temperature.
According to above-mentioned Taylor’s assumption, the
stress in the a phase is given as:

sa ¼ sa þ sal

ð21Þ

where sa is the thermal stress and sal is the athermal
stress in the a phase.
The b phase has a significant effect on the mechanical
behavior of the two-phase titanium alloys above the b
transus temperature. At very high temperature, stress is


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J. Luo et al. / Mechanics of Materials 42 (2010) 157–165

considered to be purely thermal and is produced by the
interaction of short-range obstacles to dislocation movement. Strain hardening is negligible due to intense recovery. Hence, the stress of the b phase is represented by
only one thermally activated stress component (Eq. (14)).

alloys, the transition from a phase to b phase will occur
in the high temperature deformation, therefore, Eq. (25)
is suitable for near-a and a type titanium alloys. However,
for near-b and b type titanium alloys, Eq. (25) is simplified
as:

sb ¼ sb

r ¼ Msb

ð22Þ

where sb is the thermal stress in the b phase.
By substituting Eqs. (21) and (22) into Eq. (18), the
modified rule of mixture is rewritten as:

s ¼ n1 fa sa þ n2 fb sb



¼ n1 fa ðTÞ sa þ sal þ n2 fb ðTÞsb

ð23Þ

The above discussion is at the mechanism level and
hence the relevant quantities are the shear stress and
strain. When comparing the prediction of the constitutive
model with the experimental data, the conversion from
shear stress to normal stress is performed using an average
Taylor factor M (Roters et al., 2000):

r ¼ Ms

ð24Þ

where M is the Taylor factor. For most of the engineering
materials, the Taylor factor: M = 3.06 (Stoller and Zinkle,
2000).
3.3. Microstructure-based constitutive model
In summary, a constitutive model in elevated deformation temperature of the two-phase titanium alloys is expressed as follows:

r ¼ Ms
s ¼ n1 fa sa þ n2 fb sb
fb ðTÞ ¼

T
T sus



w

fa ðTÞ þ fb ðTÞ ¼ 1

sa ¼ sa þ sal
sb ¼ sb
pffiffiffiffi
sl ¼ alðTÞb q þ Kd1=2
sa ¼ s0a

"

RT
c_ a0
1
ln
DGa
c_


1=qa

#1=pa

ð25Þ

 #1=pb
RT
c_ b0 1=qb
sb ¼ s0b 1 
ln
DGb
c_
Q dm
pffiffiffiffi p
q_ ¼ a1 qje_ j  a2 e RT q
"



Q

pd
c
c
c
d_ ¼ b0 d 0 T 1 e RT þ b1 je_ jd 1  b2 q_ c3 d 2
e
lðTÞ ¼ l0 
expðT r =TÞ  1

where DGb is 267.37 kJ/mol, Qdm is 20 kJ/mol, w is 14.78,
the b transus temperature of Ti–6Al–4V titanium alloy is
about 1263 K. For the Ti–6Al–4V titanium alloy, l0 is
49.02 GPa, e is 5.821 GPa, Tr is 181 K (Picu and Majorell,
2002), n1, n2, s0a ; s0b ; DGa ; c_ a0 ; c_ b0 , qa, qb, pa, pb, a1, a2, b0,
b1, b2, c0, c1, c2 and c3 are material constants.
The present constitutive model is universal to describe
the deformation behavior of titanium alloys in high temperature deformation. For near-a and a type titanium

"

sb ¼ s0b 1 
pffiffiffiffi

RT
c_ b0
ln
DGb
c_



1=qb #1=pb

ð26Þ

Q dm

q_ ¼ a1 qje_ p j  a2 e RT q
Q

pd
c
c
c
d_ ¼ b0 d 0 T 1 e RT þ b1 je_ jd 1  b2 q_ c3 d 2

4. Identification of material constants
The material constants within the constitutive model
are determined from the experimental data using the
GA-based objective optimization technique (Lin and Yang,
1999). Using the conventional optimization method, it is
very difficult to search for the global minimum in the
multi-modal distribution space. The GA method is a stochastic search method based on evolution and genetics,
and exploits the concept of the survival of the fittest (DeJong, 1999). For a given problem, there exists a multitude
of possible solutions that form a solution space. In GA, a
highly effective search of the solution space is performed,
allowing a population of strings representing the possible
solutions to evolve through the basic random operators of
selection, crossover, and mutation (Castro et al., 2004).
Therefore, a GA-based optimization technique is used to
determine the material constants in the constitutive
model. For the constitutive model, two objective functions are defined in terms of the square of the difference
between the experimental and the predicted data for
grain size of prior a phase and flow stress in the following form:

f1 ðxÞ ¼
f2 ðxÞ ¼

l1 X
n1 X
m1
X
k¼1 j¼1

i¼1

l2 X
n2 X
m2
X
h¼1 q¼1 p¼1

wijk

 
  2
c
e
di j  di j

wpqh

k



rcp

 
q

ð27Þ

k

h





rep

  2
q

h

ð28Þ

where f1(x) and f2(x) are the residuals for grain size and
flow stress, x (x = [x1, x2, . . ., xs]) represent the material constants, s is the number of the constants to be determined,
c
e
ððdi Þj Þk and ððdi Þj Þk are the predicted and experimental
grain size at time i, strain rate j and deformation temperac
ture k. The predicted grain size ððdi Þj Þk is obtained from the
grain growth Eq. (12) by means of a numerical integration
method, m1 is the number of the experimental average
grain size at deformation temperature k and strain rate j,
n1 is the number of strain rate, l1 is the number of deformation temperature, and wijk is the weight coefficient. Similarly, ððrcp Þq Þh and ððrep Þq Þh are the predicted and
experimental flow stresses at time p, strain rate q and
deformation temperature h. The predicted flow stresses
ððrcp Þq Þh is obtained from the flow stress Eq. (25), m2 is
the number of the experimental flow stress at deformation

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J. Luo et al. / Mechanics of Materials 42 (2010) 157–165

constants within the constitutive model is to minimize
the above objective functions.

5. Application of the constitutive model to Ti–6Al–4V
titanium alloy
5.1. Experimental procedures

Fig. 1. Optical micrograph of the as-received Ti–6Al–4V titanium alloy.

temperature h and strain rate q, n2 is the number of strain
rate, l2 is the number of deformation temperature, and wpqh
is the weight coefficient. The determination of material

The chemical composition of as-received Ti–6Al–4V
titanium alloy is composed of: 6.50Al, 4.25V, 0.16O,
0.04Fe, 0.02C, 0.015 N, 0.0018H, balance Ti. The b transus
temperature is about 1263 K. The optical micrograph of
as-received Ti–6Al–4V titanium alloy is shown in Fig. 1.
It is seen from Fig. 1 that the original microstructure consists of equiaxed a phase with an average grain size of
about 10.0 lm, secondary (platelet) a phase and a small
amount of intergranular b phase. The heat treatment prior

Table 1
Domains of the material constants in the constitutive model of Ti–6Al–4V titanium alloy.
0.1 6 a1 6 3.5
0.1  106 6 b2 6 1.0
0.5 6 c3 6 10.0

0.1  1012 6 a2 6 2.0
1.0 6 c0 6 20.0
1.0 6 n1 6 2.5

50.0 6 b0 6 5000.0
0.1 6 c1 6 4.0
1.0 6 n2 6 2.5

100:0  s0b  2000:0

100.0 6 DGa 6 700.0

1:0  c_ a0  1:0  1020
0.1 6 qa 6 20.0

0.1 6 pa 6 2.0

0.1 6 pb 6 2.0

0.1 6 b1 6 4.0
0.1  106 6 c2 6 1.0
100:0  s0a  2000:0
1:0  c_ b0  1:0  1020
0.1 6 qb 6 20.0

Table 2
Optimized material constants for the constitutive model of Ti–6Al–4V titanium alloy.

a1

a2

b0

b1

b2

c0

2.9446

1.5934  105

659.5581

0.6078

9.2954  103

11.6473

c1

c2

c3

n1

n2

s0a ðMPaÞ
1078.6451

3

0.2141

6.6181  10

1.0677

2.0865

1.6528

s0b ðMPaÞ

DGa (kJ/mol)

c_ a0 ðs1 Þ

c_ b0 ðs1 Þ

pa

pb

1327.6880

560.2436

1.0  1017

1.0  105

0.3189

0.5554

qa

qb

1.1131

13.7706

13

13

(a)

11

10

9

Average relative difference=6.19%

8

8

9

10

11

Experimental results/µm

12

(b)

11

10

9

Average relative difference=7.94%
8

7
7

Deformation temperature=1093~1243K
-1
Strain rate=0.001~10.0s
True strain=0.22~0.92

12

Calculated results/µm

Calculated results/µm

Deformation temperature=1093~1243K
-1
Strain rate=0.001~10.0s
12
True strain=0.22~0.92

13

7
7

8

9

10

11

Experimental results/µm

Fig. 2. Comparison of the predicted with the experimental grain size of prior a phase: (a) sampled; (b) non-sampled.

12

13

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J. Luo et al. / Mechanics of Materials 42 (2010) 157–165

to isothermal compression was conducted in the following
procedures: (1) heating to 1023 K and holding for 1.5 h and
(2) air-cooling to room temperature. The cylindrical compression specimens have 8.0 mm in diameter and
12.0 mm in height, and the cylinder ends were grooved
for retention of glass lubricants in the whole process of isothermal compression.
To investigate the effect of processing parameters on
deformation behavior of Ti–6Al–4V titanium alloy, a series of the isothermal compressions were conducted on a
300

Thermecmaster-Z simulator at the deformation temperatures of 1093, 1123, 1143, 1163, 1183, 1203, 1223, 1233,
1243, 1253, 1263, 1273, 1283, 1293, and 1303 K, with the
strain rates of 0.001, 0.01, 0.1, 1.0, and 10.0 s1, and the
height reductions of 20%, 30%, 40%, 50%, and 60%. The
specimens were heated and held for 3.0 min at
the each deformation temperature so as to obtain a uniform deformation temperature. After isothermal compression, the specimens were cooled in air to room
temperature.
300

Calculated
Experimental

(a)
250

10.0s

200

1.0s

Calculated
Experimental

(b)

-1

250

Flow stress/MPa

Flow stress/MPa

-1

-1

-1

0.1s

150

-1

0.01s
100

10.0s

200

-1

1.0s
150

-1

0.1s
100

-1

0.01s

-1

0.001s

50

0
0.0

0.1

0.2

-1

0.001s

50

0.3

0.4

0.5

0.6

0
0.0

0.7

0.1

0.2

0.3

Strain

0.5

300

300

Calculated
Experimental

(c)

0.6

0.7

Calculated
Experimental

(d)
250

200

10.0s

150

1.0s

100

0.1s

Flow stress/MPa

250

Flow stress/MPa

0.4

Strain

-1

-1

-1

200

10.0s

150

1.0s

-1

-1

100

0.1s

-1

-1

0.01s
50

0.01s

50

-1

0.001s

0.001s

0
0.0

0.1

0.2

-1

0.3

0.4

0.5

0.6

0
0.0

0.7

0.1

0.2

-1

0.3

300

(f)

250

0.6

0.7

Calculated
Experimental

250

Flow stress/MPa

Flow stress/MPa

0.5

300

Calculated
Experimental

(e)

200

150
-1

10.0s
100

-1

200

150
-1

10.0s

100

1.0s

-1

1.0s

-1

0.1s
50

0
0.0

0.4

Strain

Strain

0.01s
-1
0.001s
0.1

0.2

-1

0.1s
-1
0.01s
-1
0.001s

50

-1

0.3

0.4

Strain

0.5

0.6

0.7

0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Strain

Fig. 3. Comparison of the predicted with the experimental flow stress: (a) 1123 K; (b) 1143 K; (c) 1163 K; (d) 1183 K; (e) 1203 K; (f) 1223 K.

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J. Luo et al. / Mechanics of Materials 42 (2010) 157–165

In order to measure the grain size of post-compressed
specimens, the isothermally deformed specimens were
axially sectioned and prepared using standard metallographic techniques. The measurement of grain size was
carried out in an OLYMPUS PMG3 microscope with the
quantitative metallography SISC IAS V8.0 image analysis
software. And, 4 measurement points and 4 visual fields
of every measurement point in the different deformation
regions were chosen. The grain size of prior a phase was
calculated by the average values of 16 visual fields.
5.2. Determination of the material constants
The selected experimental data in the deformation temperature ranging from 1093 to 1303 K and the strain rate
ranging from 0.001 to 10.0 s1 are chosen as the sample
data to determine the material constants in the constitutive model. Other experimental data are used to verify
the model. The domains of the material constants are listed
in Table 1. Moreover, the optimized material constants
using the GA-based objective optimization technique are
listed in Table 2.

on evolution and genetics. Two objective functions
are defined in terms of the square of the difference
between the experimental and the predicted data
for average grain size of prior a phase and flow
stress.
(3) The constitutive model is applied to represent the
deformation behavior in isothermal compression of
Ti–6Al–4V titanium alloy. The average relative difference between the predicted and the experimental
flow stress is 6.13%, and those of the sampled and
the non-sampled grain size are 6.19% and 7.94%,
respectively. It can be seen that the present constitutive model with a high prediction precision can be
used to describe the deformation behavior in high
temperature deformation of titanium alloys.
Acknowledgment
The authors thank the financial supports from the fund
of the State Key Laboratory of Solidification Processing in
NWPU with Grant No. KP200905.

5.3. Comparison of the predicted with the experimental data

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Fig. 2(a) shows the comparison of the predicted with
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relative difference is 6.19%. Fig. 2(b) shows the comparison
of the predicted with the non-sampled grain size of prior a
phase, and the average relative difference is 7.94%.
The comparison of the predicted with the experimental
flow stress is showed in Fig. 3. It is seen that the average
relative error between the experimental and the predicted
flow stress is 6.13%. It can thus be concluded that the constitutive model can efficiently predict the deformation
behavior in high temperature deformation of Ti–6Al–4V
titanium alloy.

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