A physically-based constitutive model for hot deformation of Ti-10-2-3 alloy

Accepted Manuscript A physically-based constitutive model for hot deformation of Ti-10-2-3 alloy Ravindranadh Bobbili, B. Venkata Ramudu, Vemuri Madhu PII:

S0925-8388(16)33688-X

DOI:

10.1016/j.jallcom.2016.11.208

Reference:

JALCOM 39705

To appear in:

Journal of Alloys and Compounds

Received Date: 2 October 2016 Revised Date:

13 November 2016

Accepted Date: 15 November 2016

Please cite this article as: R. Bobbili, B. Venkata Ramudu, V. Madhu, A physically-based constitutive model for hot deformation of Ti-10-2-3 alloy, Journal of Alloys and Compounds (2016), doi: 10.1016/ j.jallcom.2016.11.208. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A physically-based constitutive model for hot deformation of Ti-10-2-3 alloy

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RAVINDRANADH BOBBILI*, B.VENKATA RAMUDU, VEMURI.MADHU

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Defence Metallurgical Research Laboratory, Hyderabad 500058, India

Abstract

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Hot compression tests of Ti-10-2-3 alloy were carried out in the temperature range of 90010500C, at strain rates of 0.001–1 s−1 on a Gleeble thermo mechanical-simulator. The experimental results demonstrate that the flow stress of the titanium alloy is significantly influenced by

temperature, strain and strain rate. A physically-based constitutive model was

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established, based on the dislocation density theory and dynamic recrystallization kinetics. The comparison between the experimental and predicted flow stresses shows that the established model has high accuracy. The statistical study confirmed the predicting capability of the model.

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The presented constitutive model, as well as the dynamic recrystallization (DRX) kinetics, was incorporated into ABAQUS to provide an effective means to study hot deformation. Results

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confirm that the constitutive model considering dynamic recovery (DRV) and DRX provides high accuracy. DRX results of the Finite Element Method (FEM) are in good agreement with the experimental results. Keywords:

Numerical modeling; Physically-based model; Titanium alloy.

*Corresponding author. Tel. 040 24346332; fax: 040 24342252. E-mail address: [email protected] 1

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1. Introduction The constitutive modeling of metals is a crucial step in understanding deformation behaviors of

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material during processing [1-2]. It is the basis to perform numerical simulation of metal forming process. Numerical simulation of plastic deformation processes has been an essential tool to achieve exceptional material properties and to control microstructures. Precise elucidation of flow stress is the key requirement for generating realistic numerical simulation results. During

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hot deformation process, Constitutive model denotes the complex nonlinear relation between the flow stress, strain, strain rate and temperature [3-5]. In recent times numerous constitutive

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models have been generated or modified to illustrate the hot deformation characteristics of materials. The constitutive models [6-10] are broadly split into three groups, including the phenomenological, physically-based and neural network models. Phenomenological constitutive models such as Johnson-Cook (J-C) and Zerilli-Armstrong (Z-A) model have been generally

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employed to determine flow stress under different strain rates and temperatures. The phenomenological models do not consider the effects of physical deformation mechanisms and express flow stress in a simple mathematical equation with fewer variables [11-14].

It was

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noticed that these models cannot describe the material deformation mechanisms, mainly strain hardening, DRV and DRX conditions. Based on the dislocation density evolution, various

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physical-based constitutive models have been developed to elucidate the work hardening [1517], dynamic recovery (DRV) and dynamic recrystallization (DRX) behavior of plain carbon steels, 42CrMo steel, a nitrogen alloyed

steel, a typical nickel-based superalloy and Cu-

0.4 Mg alloy [18-22]. A few studies have been made, to predict the flow behavior of titanium alloys [23-24] through a physical approach because of the material's heterogeneous deformation behavior and complex microstructural evolution. However, a combined approach to predict the

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work hardening/dynamic recovery up to the peak strain and the flow softening behavior beyond the peak strain has not yet been developed for most of the titanium alloys except Ti6Al4V and Ti-Nb-Al alloys [23-24].

The physical-based models can describe these mechanisms very

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accurately. Nevertheless, these models need large material constants under controlled experiments, through computational techniques. Therefore, it is of enormous need to establish constitutive models for this titanium Ti-10-2-3 alloy. The physically-based constitutive models

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are not available for the studied material. Ti-10-2-3 titanium alloy, is a metastable beta titanium alloy with two very important features: (1) the overall performance is superior to any other

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known titanium alloy; (2) malleability is better than that of any other known titanium alloys. In this work, high-temperature compression tests of Ti-10-2-3 alloy were carried out in the temperature range of 900-10500C, at strain rates of 0.001–1 s−1 on a Gleeble thermo mechanicalsimulator. The experimental results demonstrate that the flow stress of the titanium alloy is

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significantly influenced by temperature, strain and strain rate. A physically-based constitutive model was established, based on the dislocation density theory and dynamic recrystallization kinetics. The comparison between the experimental and predicted flow stress shows that the

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the model.

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established model has high accuracy. The statistical study confirmed the predicting capability of

2. Experimental procedures The high temperature experiments were performed in accordance with ASTM: E209 standard. In the present study a beta-titanium alloy, Ti-10V-2Fe-3Al (Ti-10-2-3) was used. The material of chemical composition (wt. %): V-9.91, Al-3.28, Fe-2.09, C-0.030, N-0.005, H-0.0014, O-0.101 and Ti (balance). The beta transformation temperature of the alloy was estimated to be 730 °C. The specimens were solutionised at 765 °C for 30 min and water quenched. The hot deformation 3

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tests were conducted on a Gleeble thermo-mechanical simulator at temperatures of 900, 950, 1000 and 10500C and at strain rates of 0.001, 0.01, 0.1 and 1 s−1. The deformed specimens were heated to the deformation temperature at a heating rate of 30 K/s and held at that temperature for

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180 s. The samples were quenched after deformation immediately with helium gas to retain the original microstructure. After polished mechanically and etched in the Kroll’s reagent, the exposed surfaces were observed by optical microscope (OM). The microstructure of the heat

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treated alloy was metallographically examined using Kroll’s reagent (6 ml HNO3/ 3 ml HF/ 100

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ml water) under optical microscope. 3. Results and discussion

3.1. High temperature deformation behavior

The true stress–strain curves for titanium alloy during high temperature deformation are demonstrated in fig. 1. For a particular temperature the flow stress will raise with rising strain

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rate, while for a particular strain rate (Fig. 1 (b)) the flow stress will reduce with increasing temperature. for a particular strain rate, the density of dislocation decreases with increasing

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temperatures, so the overall level of the flow stress will diminish with rising temperatures; similarly for a particular temperature, the density of dislocation raises with increasing strain

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rates, so the overall level of the flow stress will go up with rising strain rates. It is observed from the microstructural observation of the specimens compressed at various strain rates and temperatures, with raise in strain rate, the microstructures transform from recrystallized grains to flat grains (fig.2). At the temperature of 900 °C and the lower strain rate of 0.01 s−1, dislocation accumulation would decrease and interaction between dislocations would be insignificant, due to the annihilation and rearrangement of dislocations.

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Generally, the dynamic softening mechanisms [23-24], in the true stress–strain curves for titanium alloy can be split into two stages, i.e., DRV and DRX. It is observed from fig.1 that the flow stress rises due to work-hardening. It is attributed to the production and pile-up of

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dislocations happen immediately, leading to the high work-hardening rate [24]. When equilibrium is reached between work-hardening and DRV, a saturation flow stress stays constant. DRX occurs at a critical strain (ε c ), resulting in decreased work-hardening rate with increase in

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deformation. When the dynamic softening rate is in balance with the work-hardening rate (θ), the flow stress attains a peak stress (σ p ), and then steadily decreases to saturation stress (σ s s ). It is

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noticed by plotting peak stress, saturation stress and critical strain curves (fig.3). 3.2. Constitutive modeling of hot deformation

Constitutive equations in relation to flow stress (σ), strain (ε), strain rate ( ), and temperature (T)




Z= ε. exp ( ) 

ε. = A F (σ) exp (−






)

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[10-13, 25] have been developed to understand hot deformation behavior of the materials.

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σ ασ < 0.8 F (σ) = exp (βσ) ασ > 1.2 [sinh(ασ)] for all σ

(1)

(2)

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(3)

where ε. is the strain rate, σ is the flow stress (MPa), A (/s), α (MPa-1), β (MPa-1) and n are the coefficients relative to the material, R is the molar gas constant (8.3145Jmol−1 K−1), T is the absolute temperature( K), Zener-Hollomon parameter (Z) and Q is the hot deformation activation energy(J/mol). The flow stress varies with the strain variation during the hot deformation of titanium alloy, so the coefficients (A, α, β, n and Q) are supposed to be related with the strain.

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The values of n1 and β are determined from the relationships of lnε.-lnσ and lnε̇-σ, as plotted in fig.4. It is noticed that the flow stresses are well fitted by a series of straight lines. Accordingly, the mean values of n1 and β are calculated as 6.4 MPa−1 and 0.432. Thus, the material constant of

as the following equation by differentiating Eq. (4). & '(()*+ασ, -) . /

&( )

0

(4)

1̇

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! = #. $ %

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α can be determined as 0.067. Hence, the activation energy Q of hot deformation [25] is obtained

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Similarly, the value of ∂[lnsinh(ασ)]/∂(1/T)∂[lnsinh(ασ)]/∂(1/T) can be evaluated by fig.5 (a-b). Thus, according to Eq. (4), the activation energy Q of titanium alloy is 534 kJ/ mol. The values of material constants of n and A can be obtained by the following expression

lnZ=lnA+nln[sinh(ασ)]

(5)

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Then, by solving Eq. (5) using the definition of hyperbolic sine function, the flow stress σ can be written as a function of Z parameter. The values of Z parameter [25] under different deformation condition are determined by introducing the value of Q into Eq. (1). Fig.5 shows the relationship In this study, the calculated values of n and A are 5.4 and

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between ln Z and ln[sinh(ασ)].

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8.3 × 1020, respectively. It was often assumed that effects of the strain on flow stresses at elevated temperatures were unimportant and it was usually ignored in Eq. (1). However, it has been recently shown that the deformation activation energy Q was strongly influenced by the strain, and the effect of the strain on the material constants (i.e. a, n and lnA) was significant among the entire strain range as shown in fig. 6. Therefore, the compensation of the strain should be taken into account in order to establish the constitutive equation to predict the flow stress more accurately, and the influence of the strain in the constitutive equation was incorporated by

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assuming that the activation energy (Q) and other material constants (i.e. a, n and lnA) were polynomial functions of the strain. A forth order polynomial fitting is adopted to establish the relationship between the strain and the material constants of α, Q, n and ln A, as expressed in the

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following equation (6) Q=B0+B1ε+ B2ε2+ B3ε3+ B4ε4 n=D0+D1ε+ D2ε2+ D3ε3+ D4ε4

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(6)

α=F0+F1ε+ F2ε2+ F3ε3+ F4ε4

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ln A=E0+E1ε+ E2ε2+ E3ε3+ E4ε4

3.3. Modeling the work-hardening and dynamic recovery

During work hardening-dynamic recovery period, the evolution of dislocation density [19-20] with strain is generally considered to depend on two components: multiplication and annihilation

dρ/dε = U- Ωρ

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of dislocation, and can be expressed as

(7)

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where dρ/dε is the increase rate of dislocation density with strain; U is a multiplication term representing the work hardening. Ωρ represents the dynamic recovery due to the dislocation

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annihilation and rearrangement [21-22]. For the sake of fitting well data of the saturation stress (σsat), a third order polynomial is taken to model the relation between the measured saturation stress (σsat) and lnZ, and can be expressed as follows: σsat = 3.381 Χ 10-4 (lnZ)3+0.3486(lnZ)2-23.87 (lnZ)+409.8

7

(8)

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Fig. 7 shows that the third order polynomial for the saturation stress (σsat) greatly improves prediction accuracy of the saturation stress. The third order polynomial could well establish the

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relation between the yield stress (σ0) and lnZ and can be expressed as

σ0 = 2.51 Χ 10-4 (lnZ)3+0.2415(lnZ)2-10.24 (lnZ)+542

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(9)

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Fig. 8 shows that there exists a linear relation between ln Ω and lnZ. It is observed that the dynamic recovery coefficient increases with the decrease of Zener-Hollomon (Z) parameter, and Ω can be expressed as a function of Z parameter.

(10)

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Ω= 1245.63Z-0.0446

Therefore, the constitutive relation of titanium alloy during the work hardening-dynamic recovery period can be summarized as

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σrec = σ2sat + (σ20 -σ2sat ) 2 34Ɛ

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σsat = 3.381 Χ 10-4 (lnZ)3+0.3486(lnZ)2-23.87 (lnZ)+409.8 σ0 = 2.51 Χ 10-4 (lnZ)3+0.2415(lnZ)2-10.24 (lnZ)+542 Ω= 1245.63Z-0.0446 Ɛp= 0.024Z0.0446 Z= Ɛ ̇exp (

567899 :;

)

8

(11)

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3.4. Modeling the dynamic recrystallization

The DRX development of metals during hot deformation mainly depends on dislocation density

value, DRX nucleation would grow up near grain boundaries. Ɛ3Ɛ< n )] Ɛ>

}

(12)

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σdrx = σrec - (σsat - σss) { 1-exp[-k(

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through free energy storage. When dislocations constantly increases and attains to a critical

where εc and εp are the critical strain and peak strain, respectively. K and n are material constants.

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These values are determined as: k=1.01 and n= 0.74. Meanwhile, XDRX [11-12] can also be represented as

X DRX =

σ – @>

@A 3 σ,

(13)

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In Eq. (12), there are five unknown parameters, including εc, εp, σss, kd and nd, that needs to be determined. Fig. 9 demonstrates there exists a linear relation between ln(εp) and lnZ. So, the

(14)

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Ɛp= 0.024Z0.0446

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peak strain εp can be represented as a function of Z as follows:

So, the critical strain εc can be represented as a function of Z (fig.10) as follows:

Ɛc = -0.0002415(lnZ)2 + 0.021 (lnZ) - 0.542

(15)

In addition, the average absolute relative error (AARE) and correlation coefficient (R) are employed to assess the prediction accuracy of the proposed constitutive model. The AARE is 9

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estimated by term comparison of the relative error. The correlation coefficient (R) represents the strength of linear relation between the predicted and experimental values (fig.11). They can be expressed as

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G G ∑H IJ.(CF3C)(DF3D )

P

CF3DF

MM$N (%) = ∑Q FSP R Q

CF

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H G L G L K∑H IJ.(CF3C ) K∑IJ.(DF3D )

R

3.5. Numerical simulations

(16)

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BCD =

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In forging, the cylinder specimen is a 3D deformable solid body, so the 3D linear reduction integration continuum element with eight nodes (C3D8R) is employed. The dies are treated as rigid bodies (fig.12). In simulating the large plastic deformation process, because of the flow of

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materials, severe mesh distortion may occur, which will lead to the loss of computation precision or even termination in the simulation. This element has the ability to carry out coupled thermo-

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structural analysis. The precision of finite element model always depends on meshing. Initially, the computations were done with coarser meshing. A mesh convergence study was carried out. Later the mesh refinement was done until the solution was independent of mesh refinements and the solution could be within the range of acceptance (fig.13). A numerical analysis for the modeled DRV and DRX model was implemented using a commercial finite element (FEM) code; ABAQUS user defined material subroutine UHARD. The ABAQUS software provides the user with various user-defined subroutines to simulate the non-linear material behavior, 10

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boundary condition, and constitutive behavior precisely by coding a user subroutine. In order to verify the FE numerical model, comparisons between the experimental and simulated maximum diameters (dmax) of deformed specimens were done, as shown in fig. 14. It is observed that the

FE model can precisely simulate the hot compressive deformation.

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simulated dmax is in good agreement with experimental ones. It represents that the developed the

In the simulation method, the element is first judged by the coordinate. If the element is in the

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deformation region, then the critical strain for DRX is estimated. If the total strain of the element is greater than the critical strain, DRX occurs and the DRX grain size and fraction are calculated.

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If the total strain of the element in the deformation region is lower than the critical strain, DRX does not happen and the grain size is defined to be the initial grain size.

4. Conclusions

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In this study, hot compression tests in a wide range of temperatures (900, 950, 1000 and 1050 °C) and strain rates (0.001, 0.01, 0.1, and 1 s−1) were carried out to study the hot

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deformation behavior of Ti-10-2-3 alloy. Important conclusions are presented below. 1) To characterize the hot deformation behavior of the Ti-10-2-3 alloy, a physically-based

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constitutive model was established. The developed model was split into two parts of I. Workhardening and dynamic recovery stage and II. Dynamic recrystallization stage.

2) A physically-based constitutive model is established to determine the flow stress of Ti-10-2-3 alloy during high temperature deformation. The correlation coefficient (R) and the average absolute relative error (AARE) between the measured and predicted stresses are 0.9974 and

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3.27%, respectively, indicating good prediction capability of the proposed constitutive model for DRV and DRX of titanium alloy during high temperature deformation.

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3) Physically-based model that considers the DRV and DRX mechanisms implemented into ABAQUS User Subroutine UHARD to investigate the hot deformation behavior. This result indicates that model presents a high accuracy in describing hot deformation behavior.

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Acknowledgements

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The authors would like to thank Director, DMRL for his support and encouragement throughout this work.

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ACCEPTED MANUSCRIPT Fig.1. True stress of Ti-10-2-3 alloy at different strain rates of (a) 0.01s-1 b) 1s-1 Fig. 2 Optical microstructure of Ti-10-2-3 alloy at various temperatures and strain rates (fig. (a-e)). (Strain=0.7) Fig. 3. Relationship between θ and true stress at temperature 10000C (b) ln(strain rate) vs. ln(stress)

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Fig.4 Plot of (a) ln(strain rate) vs. stress

Fig.6 variation of material parameters with strain Fig. 7. Relationship between saturation stress and LnZ. Fig.8. Relationship between ln Ω and lnZ

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Fig. 9. Relationship between peak strain and LnZ.

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(b) ln[sinh(ασ)]. vs. ln(strain rate) (c) ln Z .vs.

Fig.5 Plot of (a) ln[sinh(ασ)].vs.1000/T ln[sinh(ασ)].

Fig. 10. Relationship between critical strain and lnZ.

Fig. 11. Comparison between predicted and experimental flow stresses at strain rates of (a) 0.001s-1 (b) 0.01s-1, (c) 1s-1. (Solid lines represent experimental flow stress and data points denote predicted flow stress).

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Fig. 12. Geometric model for FEM simulation of the hot compression Fig. 13. Equivalent stress calculated by FEM for the hot compression test

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Fig. 14. The simulated results: comparisons between the experimental and simulated maximum diameter.

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120

100

80

10000C

60

10500C

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True Stress (MPa)

9500C

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9000C

40

0 0

0.1

0.2

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20

0.3

0.4

0.5

0.6

0.7

True Strain

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(a) 250

9000C 9500C

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150

10000C

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True Stress (MPa)

200

10500C

100

50

0 0

0.1

0.2

0.3

0.4

True Strain

(b)

0.5

0.6

0.7

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1100 900

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700 ϴ

1/s

500

0.1/s

0.01/s

300

-100 20

40

60

80

100

120

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0

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0.001/s

100

True stress (MPa)

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Fig. 2. Relationship between θ and true stress at temperature 10000C

(a) 9000Cand 0.01s-1

(b) 9500Cand 1s-1

0

-1

(d) 10000Cand 0.01s-1

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(e) 10500Cand 0.01s-1

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(c) 950 Cand 0.01s

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Fig. 3 Optical microstructure of Ti-10-2-3 alloy at various temperatures and strain rates (fig. (a-e)). (Strain=0.7)

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0

950°C

-1

1000°C 1050°C

-3

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Ln (strain rate)

-2

-4 -5 -6

-8 0

50

100

150

200

(a) 1 0

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-1

-3 -4

900°C 950°C

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ln (strrain rate)

-2

-5

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-6

250

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Stress (MPa)

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-7

1000°C 1050°C

-7 -8

3

3.5

4

4.5

5

5.5

6

6.5

7

ln (stress) (MPa)

(b)

Fig.4 Plot of (a) ln(strain rate) vs. stress

(b) ln(strain rate) vs. ln(stress)

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Ln(sinhασ)

2 1

0.001/s 0.01/s

0

1/s

-1 -2 0.74

0.76

0.78

0.8

0.82

0.84

0.86

SC

1000/T

M AN U

(a) 0 -1 -2 -3 -4 -5 -6 -7 -8 -1.5

-1

-0.5

TE D

Ln (strain rate)

RI PT

0.1/s

0

0.5

1

900°C 950°C 1000°C 1050°C

1.5

2

EP

Ln(sinhασ)

AC C

(b)

(c)

Fig.5 Plot of (a) ln[sinh(ασ)].vs.1000/T ln[sinh(ασ)].

(b) ln[sinh(ασ)]. vs. ln(strain rate) (c) ln Z .vs.

ACCEPTED MANUSCRIPT 50 45 40 35 Ln A

30 y = 625x4 - 1319.4x3 + 1006.2x2 - 353.77x + 84.071

25 15 10 5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

SC

True strain

RI PT

20

M AN U

(a)

0.08

y = 8.4375x4 - 16.098x3 + 10.722x2 - 2.7979x + 0.2679

0.07

TE D

0.06

α

0.05 0.04

0.02 0.01

0.1

0.2

AC C

0

EP

0.03

0.3

0.4 True strain

(b)

0.5

0.6

0.7

0.8

ACCEPTED MANUSCRIPT 7 6 5

n

4 3 2 1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

M AN U

(c)

600 500 400

TE D

300 y = -4375x4 + 8217.6x3 - 4718.7x2 + 254.07x + 620.17

200

0 0

EP

100

0.1

0.2

AC C

Q

0.8

SC

True strain

RI PT

y = 166.67x4 - 270.37x3 + 148.33x2 - 39.296x + 10.233

0.3

0.4

0.5

0.6

True strain

(d)

Fig.6 variation of material parameters with strain

0.7

0.8

ACCEPTED MANUSCRIPT 250

150 100 50 0 30

32

34

36

38

40

42

44

46

4 3 2 1

TE D

Ln Ω

50

M AN U

Fig. 7. Relationship between saturation stress and LnZ.

48

SC

Ln Z

RI PT

Saturation stress

200

0

-2 32

34

AC C

30

EP

-1

36

38

40 Ln Z

Fig.8. Relationship between ln Ω and lnZ

42

44

46

48

50

ACCEPTED MANUSCRIPT 0.5 0.45 0.4 0.3 0.25 0.2 0.15 0.1 0.05 0 32

34

36

38

40

42

Ln Z

0.4 0.35

0.15 0.1 0.05 0

48

50

TE D

0.2

EP

Critical strain

0.3 0.25

46

M AN U

Fig. 9. Relationship between peak strain and LnZ.

44

SC

30

RI PT

Peak strain

0.35

AC C

30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 Ln Z

Fig. 10. Relationship between critical strain and lnZ.

ACCEPTED MANUSCRIPT 70

9000C 60

9500C 10000C 40

RI PT

10500C

30 20

0 0

0.1

0.2

0.3

0.4

(a) 120

100

0.5

0.6

0.7

M AN U

True Strain

SC

10

TE D

9000C

80

9500C 10000C 10500C

EP

60

40

AC C

True Stress (MPa)

True Stress (MPa)

50

20

0

0

0.1

0.2

0.3

0.4

True Strain

(b)

0.5

0.6

0.7

ACCEPTED MANUSCRIPT 250

9000C

200

10500C 100

50

0 0

0.1

0.2

0.3

0.4

(c)

0.6

M AN U

True Strain

0.5

RI PT

10000C

150

SC

True Stress (MPa)

9500C

0.7

AC C

EP

TE D

Fig. 11. Comparison between predicted and experimental flow stresses at strain rates of (a) 0.001s-1 (b) 0.01s-1, (c) 1s-1. (Solid lines represent experimental flow stress and data points denote predicted flow stress).

Fig. 12. Geometric model for FEM simulation of the hot compression.

AC C

EP

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. 13. Equivalent stress calculated by FEM for the hot compression test

TE D

20

19 18.5

EP

18 17.5 17

AC C

Simulated max dia (mm)

19.5

16.5

16

16

16.5

17

17.5

18

18.5

19

19.5

20

Experimental max dia (mm)

Fig. 14. The simulated results: comparisons between the experimental and simulated maximum diameter.

ACCEPTED MANUSCRIPT

Highlights

The hot deformation behavior of a Ti-10-2-3 alloy is studied.

•

A physically-based constitutive model is established for the titanium alloy.

•

The predicted results by model agree well with experimental ones.

•

DRX results of the Finite Element Method (FEM) are matching with the experimental results.

AC C

EP

TE D

M AN U

SC

RI PT

•