Constitutive model for high temperature

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 (deforma-tion temperature, strain rate and strain) are established using the disloca(deforma-tion density rate as an internal state variable. Secondly the flow stress model in high temperature deforma-tion 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-param-eter internal state variables. Finally, a constitutive model coupling the grain size, volume fraction and dislocation density is developed based on the microstructure and crystal plas-ticity 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 prioraphase 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 stiff-ness 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

material composition in the forming process. Therefore, reliable constitutive equation, which describes the correla-tion of dynamic material performance with processing parameters, needs to be developed in order to understand the deformation behavior in depth and optimize the defor-mation 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 Mad-din, 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 mod-els were mostly obtained. But, those regression modmod-els are unsatisfied because the complicated and non-linear

0167-6636/$ - see front matterÓ2010 Published by Elsevier Ltd. doi:10.1016/j.mechmat.2009.10.004

*Corresponding author. Tel.: +86 29 88491478. E-mail address:honeymli@nwpu.edu.cn(M. Li).

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Mechanics of Materials

relationships may exist between flow stress, microstruc-ture and processing parameters in the high temperamicrostruc-ture deformation. The artificial neural network (ANN) method unlike the regression method, however, does not need a mathematical formulation and has the capability of self-organization or ‘‘learning”. This approach is especially suit-able for treating non-linear phenomena and complex rela-tionships 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 mech-anism-based model which focused on the low temperature material behavior and considered the deformation to be controlled by thermally activated process only. For a re-view of more recent development in these models refer toDonahue et al. (2000), Picu and Majorell (2002), Kim et al. (2003, 2005) and Voyiadjis and Almasri (2008). In re-cent 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) pre-sented a single internal state variable constitutive model to represent the deformation behavior of metals that exhib-ited flow softening caused by the competing hardening and recovery processes and heat generation during plastic deformation.Roters et al. (2000)introduced a new work-hardening 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 relation-ships 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 dis-location 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 Sher-cliff, 2002).

For the internal state variable-based constitutive equa-tion, 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 proper-ties of material. Therefore, an appropriate representation of mechanical behavior has to be based on the microstruc-tural 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; Es-trin et al., 1998; Fedelich, 1999; Goerdeler and Gottstein, 2001; Karhausen and Roters, 2002; Ganapathysubramani-an Ganapathysubramani-and Zabaras, 2004; Lin Ganapathysubramani-and DeGanapathysubramani-an, 2005; Beyerlein Ganapathysubramani-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 process-ing 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 estab-lished 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 microstruc-ture evolution on the viscoplasticity of material are analyzed. And, a constitutive equation, in which the dislo-cation density and grain size of matrix phase are taken as internal state variables, is developed based on the physi-cally microstructure model. Applying these models to the isothermal compression process of Ti–6Al–4V titanium alloy, the material constants in these models are deter-mined by a genetic algorithm (GA)-based objective optimi-zation technique.

2. Microstructure model

2.1. Dislocation density rate

It is well known that dislocation density

q

in high tem-perature deformation of metals and alloys depends on two competing processes: working hardening and dynamic softening.Mecking and Kocks (1981)had pursued a phe-nomenological 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 repre-sents the entire current structure (Ding and Guo, 2002). In the KM model, the dislocation storage rate is propor-tional to

q

1/2_{, and the dislocation annihilation rate is}

pro-portional to

q

, so the variation of dislocation density with strain can be written as:

d

q

d

e

p¼k1

ffiffiffiffi

q

p _k

2

q

ð1Þ

where

q

is the average dislocation density,

e

p_{is the plastic}

strain,k1is a material constant and describes the storage of

dislocations at the dislocation forest obstacles,k2describes

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

k2¼k20

e

_

whereR is the gas constant (8.314 J/(mol K)),Qdmis the

activation energy for cross-slip and recombination (kJ/ mol),k20is a proportional constant,

e

_0and

e

_are a reference

and the applied strain rate, andRcis a cut-off radius

be-yond 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 ther-mally 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 dis-location density

q

is so large that e0:7R4

cq2in Eq.(2)is close

to zero. Therefore, Eq. (2) is simplified to the following equation:

Combining Eqs.(1) and (3), the present dislocation den-sity rate

q

_ in high temperature deformation can be ex-pressed as:

right hand side of Eq.(4)has two terms, the first one char-acterizes the processes of dislocation storage, and the sec-ond characterizes the concurrent dislocation annihilation by dynamic recovery.

2.2. Grain growth rate

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 plas-tic strain, and grain growth due to the variation of disloca-tion density (Luo et al., 2008).

The static grain growth as an atomic diffuse process af-fected by thermal effect relates to grain boundary mobility. Therefore, the static grain growth is given by the following expression (Shewmon, 1969):

_

dstatic¼

M

r

surf

d ð5Þ

whered_staticis the static grain growth rate,dis the average

grain size (

l

m),

r

surfis the grain boundary energy per unit

area (J/m2_{), and}

Mis the grain boundary mobility (m4_{/(J s)),}

which can be written as (Ding and Guo, 2002):

M_¼dDb

whereDis the boundary self-diffusion coefficient (m2_/s),

which is an exponential function of deformation tempera-ture.dis the characteristic grain boundary thickness (m),

bis the Burgers vector magnitude (2.861010_m),D 0is

the self-diffusion constant, k is Boltzmann’s constant (1.3811023_{J/K), andQ}

pdis the boundary diffusion

acti-vation energy (kJ/mol). In Eq.(6)the effect of the exponen-tial term is larger than that of 1/T, hence the grain boundary mobility increases with an increase of deforma-tion temperature.

Substituting Eq.(6)into Eq.(5), the static grain growth

rate_d__staticcan be written as:

_

dstatic¼b0d

c0

T1_eQpdRT ð7Þ

where

c

0andb0(dD0

r

surfb/k) are the

temperature-depen-dent material constants. The right hand side of Eq.(7) re-flects temperature dependence of the static grain growth. In high temperature deformation, the dynamic grain growth induced by plastic strain can be expressed as ( Dun-ne, 1998; Zhou and DunDun-ne, 1996):

_

ddyn¼b1j

e

_jd

c1

ð8_Þ

whereb1and

c

1are the temperature-dependent material

constants.

The effect of dislocation density on grain size is consid-ered in high temperature deformation of titanium alloys, therefore, the average grain size can be described by the following equation (Li and Li, 2005):

_

ddis¼ b2

q

_c3d

c2

ð9_Þ

whereb2,

c

2and

c

3are 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 dis-location density on grain growth, the grain growth rate may be taken the form of

_

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

_

In summary, a microstructure model of metals and al-loys in high temperature deformation, with the dislocation density rate being an internal state variable, can be ex-pressed as:

3. Physical-based constitutive model

3.1. Plastic deformation mechanisms

com-puted using Taylor’s assumption by which the different mechanisms act in parallel, contributing to stress sepa-rately. The total stress is assumed to be composed of a thermally activated stress

s

* and an athermal stress

s

l

(Picu and Majorell, 2002).

s

_¼

s

_þ

_s

l 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 ele-ments, and solute atoms (interstitial and substitutional). The athermal stress

s

lis mainly due to the long-range

ef-fects 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

_¼

_s

0 ₁

_D

RT_Gln

c

_0

_

c

1=q

" #1=p

; ð14_Þ

where DG is the activation energy for deformation (kJ/ mol),

s

0

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, qand

c

_0are material constants. The

reference rate

c

_0is related to the vibrational frequency of

dislocations arrested at an obstacle, or the obstacle over-coming attempt frequency (Kocks et al., 1975).

While traditionally the thermally activated stress cap-tures 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 sizedare treated as internal state variables. The athermal stress is given by:

s

l¼a

l

ðTÞbp

q

ffiffiffiffiþKd1=2 ð15Þ

where,bis the Burgers vector magnitude (2.861010_m),

aandKare material constants,

q

is the dislocation density anddis the average grain size (

l

m). The second term in Eq.

(15)is the standard Hall Petch term which stands for the strengthening effect due to grain boundaries. The constant

Kis 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):

l

ðTÞ ¼

l

0 e

exp_ðTr=TÞ 1 ð

16Þ

whereeandTrare material constants.

In the earlier researches about constitutive model of plastic deformation, the average grain sizedis 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 tem-perature deformation. Therefore, the effect of microstruc-ture evolution on macroscopic deformation behavior in the present study is adopted and the average grain sized

is treated as an internal state variable in Eq.(15). The grain size varies with the processing parameters and is deter-mined through Eq.(12).

3.2. Constitutive model for two-phase titanium alloy

For two-phase titanium alloys, the microstructure be-low thebtransus temperature consists of

a

phase (hexag-onal close-packed, hcp) andbphase (body-centered cubic, bcc). When the grain sizes of

a

phase andbphase are in the same order of magnitude, moreover

a

phase andbphase 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 macro-scopic 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

_¼fa

s

aþfb

s

b ð17Þ

where fa and fb are the temperature-dependent volume

fractions of

a

phase and b phase, respectively, and

fa+fb= 1,

s

aand

s

bare the stresses of

a

phase andbphase,

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 ba-sis of an ideal assumption of two aligned continuous phases under iso-strain conditions. Strengthening exceed-ing the ROM averages has been observed in the deforma-tion of in situ Cu/M composites (Funkenbusch and Courtney, 1985). This has been attributed in part to the dif-ficulties 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

¼n1fa

s

aþn2fb

s

b ð18Þ

wheren1andn2are the strengthening coefficients that are

greater than 1. Finally, the variation of thebphase volume fractionfb(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:

fbðTÞ ¼ TTsus

w

T<Tsus

fa¼1fb

8

<

:

ð19_Þ

fb¼1 TTsus

fa¼0

ð20_Þ

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:

s

a¼

s

aþ

s

al ð21Þ

where

s

a is the thermal stress and

s

al is the athermal

stress in the

a

phase.

Thebphase has a significant effect on the mechanical behavior of the two-phase titanium alloys above the b

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

s

b¼

s

b ð22Þ

where

s

bis the thermal stress in thebphase.

By substituting Eqs. (21) and (22) into Eq. (18), the modified rule of mixture is rewritten as:

s

_¼n1fa

s

aþn2fb

s

b¼n1faðTÞ

s

aþ

s

al þn2fbðTÞ

s

b 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 factorM(Roters et al., 2000):

r

¼M

s

ð24Þ

whereMis 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 deforma-tion temperature of the two-phase titanium alloys is ex-pressed as follows:

thebtransus temperature of Ti–6Al–4V titanium alloy is about 1263 K. For the Ti–6Al–4V titanium alloy,

l

0 is

49.02 GPa, eis 5.821 GPa,Tris 181 K (Picu and Majorell, 2002), n1, n2,

s

0a;

s

b0;DGa;

c

_a0;

c

_b0,qa,qb,pa, pb,

a

1,

a

2,b0, b1,b2,

c

0,

c

1,

c

2and

c

3are material constants.

The present constitutive model is universal to describe the deformation behavior of titanium alloys in high tem-perature deformation. For near-

a

and

a

type titanium

alloys, the transition from

a

phase tobphase will occur in the high temperature deformation, therefore, Eq.(25)

is suitable for near-

a

and

a

type titanium alloys. However, for near-bandbtype titanium alloys, Eq.(25)is simplified as:

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 sto-chastic search method based on evolution and genetics, and exploits the concept of the survival of the fittest ( De-Jong, 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 func-tions 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 follow-ing form:

stants,sis the number of the constants to be determined,

ððdciÞjÞk and ððd e

iÞjÞk are the predicted and experimental

grain size at timei, strain ratejand deformation tempera-turek. The predicted grain sizeððdc_iÞjÞkis 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 temperaturekand strain ratej,

n1is the number of strain rate,l1is the number of

deforma-tion temperature, andwijkis the weight coefficient.

Simi-larly, _ðð

r

c

pÞqÞh and ðð

r

epÞqÞh are the predicted and

experimental flow stresses at time p, strain rate q and deformation temperature h. The predicted flow stresses

ðð

r

c

pÞqÞh is obtained from the flow stress Eq. (25), m2 is

temperaturehand strain rateq,n2is the number of strain

rate,l2is the number of deformation temperature, andwpqh

is the weight coefficient. The determination of material

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

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. Thebtransus temperature is about 1263 K. The optical micrograph of as-received Ti–6Al–4V titanium alloy is shown in Fig. 1. It is seen fromFig. 1that the original microstructure con-sists of equiaxed

a

phase with an average grain size of about 10.0

l

m, secondary (platelet)

a

phase and a small amount of intergranularbphase. The heat treatment prior

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

Table 1

Domains of the material constants in the constitutive model of Ti–6Al–4V titanium alloy.

0.16a163.5 0.11012_6a

262.0 50.06b065000.0 0.16b164.0

0.1106_{6b261.0 1.06c}

0620.0 0.16c164.0 0.1106_6c

261.0

0.56c3610.0 1.06n162.5 1.06n262.5 100:0s0

a2000:0

100:0s0

b2000:0 100.06DGa6700.0 1:0c_a01:010

20

1:0c_b01:010 20

0.16pa62.0 0.16pb62.0 0.16qa620.0 0.16qb620.0

Table 2

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

a1 a2 b0 b1 b2 c0

2.9446 1.5934105 _{659.5581 0.6078 9.295410}3 _11.6473

c1 c2 c3 n1 n2 s0aðMPaÞ

0.2141 6.6181103 _{1.0677 2.0865 1.6528 1078.6451}

s0

bðMPaÞ DGa(kJ/mol) c_a0ðs

1_{Þ c_}

b0ðs

1_{Þ pa p}

b

1327.6880 560.2436 1.01017 _1.0105 _{0.3189 0.5554}

qa qb

1.1131 13.7706

7 8 9 10 11 12 13

Average relative difference=6.19%

Calculated results/

µ

m

Experimental results/µm

(a)

Deformation temperature=1093~1243K

Strain rate=0.001~10.0s-1

True strain=0.22~0.92

7 8 9 10 11 12 13 77 8 9 10 11 12 13

8 9 10 11 12 13

Deformation temperature=1093~1243K Strain rate=0.001~10.0s-1

True strain=0.22~0.92

Average relative difference=7.94%

Calculated results/

µ

m

Experimental results/µm

(b)

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 com-pression 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 iso-thermal compression.

To investigate the effect of processing parameters on deformation behavior of Ti–6Al–4V titanium alloy, a ser-ies of the isothermal compressions were conducted on a

Thermecmaster-Z simulator at the deformation tempera-tures 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 uni-form deuni-formation temperature. After isothermal com-pression, the specimens were cooled in air to room temperature.

(a)

_ExperimentalCalculated

0

In order to measure the grain size of post-compressed specimens, the isothermally deformed specimens were axially sectioned and prepared using standard metallo-graphic 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 tem-perature 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 constitu-tive 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 inTable 2.

5.3. Comparison of the predicted with the experimental data

Fig. 2(a) shows the comparison of the predicted with the sampled grain size of prior

a

phase, and the average 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 inFig. 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 con-stitutive model can efficiently predict the deformation behavior in high temperature deformation of Ti–6Al–4V titanium alloy.

6. Conclusions

(1) A physically based constitutive model is proposed to represent the deformation behavior of titanium alloys. In the constitutive model, the total stress is assumed to be composed of a thermally activated stress and an athermal stress, in which the thermally activated stress is described by a Kock–Mecking model. The athermal stress associated with the hardening effect is represented by two-parameter internal state variables, including the dislocation density rate and the grain size rate. The role of the

a

phase andbphase on the flow stress is character-ized with the rule of mixture and superposition the-ory. The present constitutive model represents a more realistic deformation behavior for titanium alloys.

(2) A GA-based objective optimization technique has been developed to identify the material constants in the present constitutive model. GA-based optimi-zation technique can effectively solve strongly non-linear objective functions for optimization based

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 dif-ference 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 constitu-tive 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.

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