Phenomenological model of hardening and flow for Ti-6Al-4 V titanium alloy sheets under hot forming conditions

Hardening is the core factor to determine deformation uniformity in sheet metal forming. Hot deformation of titanium alloy sheets encounters the coupled effects of strain hardening, strain rate hardening and softening, which makes the determination of forming parameters aiming for an enhanced hardening very difficult in practical processes. This paper presents a new model to quantify the hardening of Ti-6Al-4 V titanium alloy sheets under hot forming conditions based on the underlying correlation between uniform strain and hardening. Firstly, to precisely determine hot deformation characteristics of titanium alloy sheets, hot tensile uniaxial tests using Gleeble systems at various strain rates of 0.01–1 s−1 and temperatures of 973–1123 K were performed systematically. A newly developed volume-based correction method for stress–strain curves of Gleeble thermo-mechanical testing was proposed to eliminate the damaging effect of temperature gradients on strain calculations, which enables the strain hardening, strain rate hardening and softening to be determined precisely. Then, a simple unified formula of hardening components (n, m and s) was proposed to predict the achievable uniform strain at certain conditions efficiently. Using which, occupation of each hardening can be quantified and compared to facilitate the determination of process parameters. Finally, a phenomenological model based on the hardening and softening components was developed to predict the hot flow behaviour. The proposed quantitative model can provide an efficient and useful approach for process designers to design process parameters driven by the objective of enhancing hardening to maximize uniform deformation during hot forming of titanium alloy sheets.


Nomenclature
S 0 NU (mm 2 ) Initial cross-section area of non-uniform temperature zone L 0 NU (mm) Initial length of non-uniform temperature zone S i NU (mm 2 ) Instant interface area of between uniform and non-uniform temperature zones L i NU (mm) Instant length of non-uniform temperature zone S I NU (mm 2 ) Deformed interface area of between uniform and non-uniform temperature zones

Introduction
Complex-shaped titanium alloy thin-walled panel structures are the critical components for high-temperature applications in aerospace and aircraft vehicles. Such titanium alloy sheet components are difficult to form using cold forming due to the poor ductility, high strength and severe springback [1,2]. Superplastic forming is a robust technology for forming extremely complex-shaped titanium alloy components, while localised thinning and strength deterioration are two dominant disadvantages need to be solved [3,4].
To address the difficulties of above-mentioned processes, novel processes of hot gas pressure forming or hot stamping have been proposed, which are performed in a middle range temperature (0.4 ~ 0.6 T m ) and rapid strain rate (~ 0.1 s −1 ). Using which, the hardening of material in these processes is not only dominated by strain rate hardening, but also strain hardening [5], which overcomes the limitation of single hardening of either cold or superplastic forming. Thus, it opens the possibility of cooperatively utilise both hardening mechanisms to improve the uniform deformation capacity of titanium alloy sheets during heat forming. The core effect of hardening on forming is the guarantee the uniformity of deformation. Significant researches have been carried out for the quantification of hardening effect on deformation uniformity at room temperature. Considère has proposed a criterion of strain hardening on the uniform strain [6]. Thereafter Hart has extended Considère criterion and incorporated strain rate hardening [7]. It is proved the strain rate hardening contributed to delay necking and improve the uniform deformation ability of the sheet metal by theoretical calculation. Ghosh et al. argues that the strain hardening exponent (n) is the most important factor to the uniform deformation of sheet metal before diffuse necking [8]. After which, the presence of strain rate sensitivity exponent (m) has an important influence on the amount of non-uniform plastic deformation. Wang et al. verified such a finding for low carbon steel as well [9]. Combination of strain hardening exponent (n) and strain rate hardening exponent (m) enables a high fracture strain indicating excellent formability. However, the above research was established on cold forming conditions, in which dislocation-based monotonous hardening dominates. For middle temperature range and relatively high strain rate forming conditions, several research has identified the beneficial effect of quick loading on the deformation uniformity [10,11], which is attributed to the cooperative function of strain and strain rate dual hardening on the enhancement of overall hardening, the researches were still limited to qualitative study without effective quantification.
For titanium alloy sheets deformed at a middle range temperature and rapid strain rate, the underlying mechanism of strain hardening is the dislocation density accumulation reflected by the flow stress increase macroscopically. Thermal activation is believed to be dominant reason for the rate sensitivity during hot forming in a middle temperature range and rapid strain rate [12]. Conventional superplastic forming of titanium alloys is dominated by single strain rate hardening, the contribution of single hardening to uniform deformation can be easily quantified, such as Alabort et al. Studied the superplastic behaviour of Ti-6Al-4 V, charted the contour map of the strain rate sensitivity exponent (m) as a function of temperature and strain rate using stress-relaxation tests and used to adjust process parameters accordingly for the manufacture of fan blade [13]. However, the quantification becomes difficult once strain hardening is introduced regarding visco-plaistc deformation, for middle temperature range and rapid strain rates. Zhu et al. studied the hot flow behaviour of three typical titanium alloys and the strain rate sensitivity exponents (n) and strain hardening exponents (m) of the three typical titanium alloys were calculated under different process parameters [14]. Considering the two components, n = ln ∕ ln and m = ln ∕ lṅ , are not at same scale, direct comparison is unable to be achieved. Then, it is difficult to know which hardening dominants at a given forming condition, and unable to adjust process parameters accordingly. In addition, for hot forming titanium alloy sheets, the dynamic softening due to void damage, grain boundary slip, deformation heat, and dynamic recrystallization normally occurs [15], which makes the above difficulty in quantification more complicated. Therefore, with an efficient and straightforward model for mathematically describing coupled hardening and softening of titanium alloy sheets of process, variable control to maximize the deformation uniformity is urgently required.
This paper presents a new model for the quantification of hardening in hot forming titanium alloy sheets. The significance of this quantification model was able to distinguish individual contribution of engineering favored n, m values on the overall hardening from the view of uniform deformation. In addition, the hot flow behaviours were precisely characterized using a volume-consistency based correction method of Gleeble thermo-mechanical testing results, which facilities the quantification of hardening. Finally, a new phenomenological model based on hardening components was established to be able to predict the hot flow behaviour and facilitate the determination of process parameters for process designers.

Material and equipment
The raw material used in the experiments was Ti-6Al-4 V (TC4) sheet with a thickness of 1.5 mm. The main chemical compositions in weight percentage are given in Table 1.
Regarding the mechanical property of as-received material, the yield strength and ultimate tensile strength are 1019.4 and 1058.2 MPa, respectively, which can be determined according to the engineering stress-strain curve at room temperature given in Fig. 1.
In order to precisely determine the hot flow and subsequent hardening behaviours, the specimen is required to be uniaxially stretched at a fixed true strain rate, while conventional material tester with furnace is normally performed at a given tension speed (mm/s), using which, the engineering strain rate is fixed and true strain rate varies. The results would be acceptable for cold forming those are nearly rateindependent. In this study, the advanced thermal-mechanical material simulator Gleeble 3800 was used, by which, the stroke versus time can be programmed to guarantee a fixed true strain rate. The temperature of specimen is able to be measured and adjusted instantly using the feedback of K-type thermocouple measurement. The temperature control precision is less than 3 K during the thermal cycle. For the uniaxial tensile test, the specimens were machined from the rolled sheets of same batch with the longitudinal direction at the rolling direction. According to Shao's experiments [16], the dimensions are shown in Fig. 2. Figure 3 shows the temperature profile of hot uniaxial tensile test, to avoid the over-shooting of specimen temperature during heating, a two-step heating strategy was used in this test, with a heating rate of 20 K/s in the first step. The heating rate in the second step was reduced to 5 K/s at a value about 50 K below the specified deformation temperature. After heating, the specimens were soaked at the target temperature for 60 s to guarantee a uniform temperature. Then, it was stretched uniaxially at a given strain rate. The temperatures of hot tensile tests were designed at 973, 1023, 1073 and 1123 K, which are within the optimum forming temperature range  The strain rate range of 0.01-1 s −1 enabled the rate range of non-superplastic forming processes to be covered [10,11]. Orthogonal experimental was used to replace conventional design of experiment approach to reduce cost and machining time for the study. In addition, a series of interrupted tests were performed, which were used to determine the deformation scenario at an instant moment during hot tension.

Test programme
3 The volume consistency-based correction model of hot uniaxial tensile tests

Method scheme and temperature characterization
For Gleeble thermo-mechanical testing, the main disadvantage is the non-uniform temperature distribution within the length of specimen due to the cooling at the electrodes. To determine the strain during testing, C-gauge can be used to measure the deformation in the width direction. Then, the tensile strain is converted based on the volume consistency principle assuming the Possion ratio is given as 0.5 [17]. However, the strain measured by C-gauge is local at the centre which is unable to indicate the uniform deformation for a large gauge zone, over-estimated deformation is normally obtained, which cannot accurately indicate the overall deformation within the uniform temperature zone. In many literatures using Gleeble testing, the stroke of grips was regarded as the displacement of the uniform temperature zone [18,19]. This method is obviously inaccurate while the measurement error becomes significantly greater once the specimen is hot. The reason is that the material outside the uniform temperature zone would be also stretched simultaneously, while this amount deformation is wrongly considered once the stroke-force method is adopted. In this research, to compensate the effect of deformation within the non-uniform temperature zone on strain calculation, a volume-consistency based correction method was proposed. The principle of this correction method is using the length of the uniform temperature zone as gauge length to calculate the strain on the premise of accurately measuring the length of uniform temperature zone. The length of the uniform temperature zone is equal to the length of the parallel zone of a Gleeble tensile specimen minus the length of the two non-uniform temperature zones. Therefore, uniform temperature zone needs to be characterized first. Figure 4 shows the schematic of the zoning of parallel zone of a Gleeble tensile specimen. In general, the parallel zone can be divided into one uniform temperature zone ( L U i ) and two non-uniform temperature zone ( L i NU ), respectively. Temperature distribution within the parallel length of specimen is characterised preliminarily using the bisection method. During testing, one thermocouple (TC1) was always welded in the middle of the sample ( T 0 ), and the other thermocouple (TC2) was welded at the different temperature measurement points, with the temperature designated as T i . i represents the index of temperature measurement points.
Selecting a temperature of 1073 K and strain rate of 0.1 s −1 , during the complete testing, including Heating I & II, soaking and uniaxial tension. The temperature difference of two positions is defined as (T 0 − T i )∕T 0 and given in Fig. 5, where the red indicates no temperature difference between two thermocouples, while the green indicates that the temperature of the thermocouple (TC1) in the middle is higher than the temperature of the thermocouple (TC2) in Fig. 4 Illustration of the nonuniformity of temperature of electrical-resistance heated specimen the side of tensile sample. As can be seen in this figure, with the position on the specimen moving towards the grip, i.e. the index increases from 1 to 5, the temperature deviations becomes greater at the stage of soaking and uniaxial tension. When the temperature deviations are less than 5% at the stage of uniaxial tension regarded as temperature distributing is evenly. In this study, the uniform temperature zone is regarded as the twice length between point 0 and 3, with a length of 30 mm. Figure 6 shows the schematic geometrical model of nonuniform temperature zone deformation. Cross-section A (red) is the interface between uniform and non-uniform temperature. Cross-section B (blue) is the base of nonuniform temperature assuming no contraction in width occurs. The region between A and B is the non-uniform temperature zone. Initially, the geometry of non-uniform temperature zone is assumed as a cuboid in Fig. 6b. Initially, before uniaxial tension, the areas of uniform temperature zone and non-uniform temperature zone at the interface are the same; thus, the area is S 0 NU , and length of non-uniform temperature zone is L 0 NU . When deformation occurs, the shape of the area deviates from cuboid. In this correction method, the non-uniform temperature zone is simplified as a hexahedron pyramid after being stretched. Therefore, the deformation of non-uniform temperature zone is the process of transforming from a cuboid to a hexahedron pyramid. The proposed correction model deals with the effect of shape variation during hot stretching on the resulted stress-strain curves. The actual shape variations were measured from the hot stretched and cooled specimen, the thermal expansion during warming is excluded, therefore this assumption only applies to the isothermal deformation process. For an instant deformation moment ( i ), the hexahedron pyramid represents the final shape, where the interface area of between uniform and non-uniform temperature zones is S I NU , and the length of deformed non-uniform temperature zone is L I NU . Considering the volume of non-uniform temperature zone is unchanged before and after deformation, thus the volume consistency equation could be established, and given as Eq. (1). In this equation, the volume of initial cuboid (left of Eq. (1)) is equal to the volume of hexahedron pyramid (right of Eq. (1)).

Mathematical model of correction method
where the instantaneous interface area between uniform and non-uniform temperature zones is S i NU , and the instantaneous length of non-uniform temperature zone is. L i NU After substitution, L i NU can be calculated as follows: The uniform temperature zone section (section A) area is assumed to experience a continuous reduction. The instantaneous area of interface boundary S i NU can be formulated as: where, I is the total number of time intervals for the deformation stage in data acquisition, i represents an instantaneous time interval.
Once the deformation of non-uniform zone is corrected, the true stress-strain curves within the uniform zone can be determined. The instantaneous gauge length L i U , i.e. the uniform temperature zone can be formulated as: where, L i is the instantaneous length of parallel zone of the specimen. The instantaneous true stress i can be calculated by: where, P i is the instantaneous deformation force, S 0 is the thickness of the test specimen, which can be manually measured from the test specimen, L 0 U is the initial gauge length with a magnitude of 30 mm in this study.
The instantaneous true strain i can be calculated by:  Fig. 7a. Figure 7b shows the true stress-strain curves of the TC4 alloy tested at temperature 1023 K and strain rate 0.01 s −1 before and after correction at gauge length, where solid symbols are experimental results, and the solid lines are

Verification of correction model
the model predictions. It can be seen that, with the increase of strain, and there exists a certain deviation between the uncorrected and corrected true stress-strain curves and the true strain for the corrected curve is reduced. The maximum value of strain correction is 0.04.

Uniform strain prediction model
For hot forming of titanium alloy sheets within the range of middle temperature and relatively high strain rates, strain hardening, strain rate hardening and softening coexist, in this section, a model for predicting the uniform strain based on the above three variables was proposed. The uniform deformation of sheet metal under specific forming conditions could be predicted through the determination of hardening components of particular stress-strain curves.
The historically first plastic instability model for the uniaxial tension of rods was proposed by Considère [6]. When plastic deformation occurs the effect of hardening on traction force is more important than a cross-section reduction, accordingly, it is assumed that the hardening cannot compensate effect of the traction force reduction due to decrease the cross-section. Corresponding to this assumption the onset of diffuse necking at the point where the rate of strain hardening ( d ∕d ) drops below the value of the flow stress at a given plastic strain rate ̇ as show in Fig. 8, which is expressed by the condition in Eq. (7), When using the Hollomon equation to describe the true stress vs. true strain relation for ductile structural materials, i.e., = K n , together with Eq. (7), the strain at which diffused necking sets in is identical with the strain exponent in the Hollomon relation: where DN is the strain corresponding to the diffuse necking point. Strain continues to be localized and the deformation often becomes intensively concentrated into a narrow band, with eventual formation of localized necking when d ∕d = ∕2 , Hill deduced this formulas from solid mechanics in 1952 [20]. Localized necking is based on zero strain along the necking band so that the onset of localized necking corresponds to a plane strain deformation mode [21]. On the other hand, it is one of the most frequently observed failure mechanisms in many sheet forming processes. The onset of localized necking limits the sheet metal formability so that determine the strain of localized necking which is of important value in evaluating the formability of sheet metals [22]. For the above-mentioned reasons Hill models is widely used in predict uniform strain of sheet. The strain ( LN ) at this point can be expressed as: When deformation passes through the load maximum, i.e. initiation of diffuse necking, the relatively gradients in strain and strain-rate in a tensile specimen begin to grow. Beyond this point, strain rate hardening occurs at different point in the diffuse necking zone of a tensile specimen. The rate of stability loss is thereby reduced and local necking condition is not satisfied until strains are greater than 2n. On this foundation, the strain rate sensitivity of the flow stress is considered by Hart, and arrived at the following criterion that includes both the strain hardening exponent and the strain rate sensitivity exponent (m): For Eq. (10), it means that the strain hardening exponent (n) and the strain rate sensitivity exponent (m) is educed, and then the strain of localized necking can be calculated. Especially for hot deformation, the softening effects due to dynamic recovery and dynamic recrystallization will promote the occurrence of plastic instability, decrease the capability of uniform deformation, and cause the localized necking occur prematurely as show in Fig. 8. In sheet metal forming diffuse and localized necking are distinguished and the maximum allowable straining is not determined by diffuse but by localized necking, therefore the deformation before localized necking can be regarded as homogeneous deformation [23]. Inspired by Hill and Hart, we established a phenomenological constitutive model to predict the uniform strain of the sheet during thermal deformation ( u ), provide references for engineering applications.
In the above equation, a softening item s is introduced to reflect the softening caused by dynamic recovery and dynamic recrystallization. From engineering practice, the strain hardening exponent (n), the strain rate sensitivity exponent (m) and the softening exponent (s) can be calculated by: The value of strain hardening item (n) can be obtained from the slope of the line in the ln − ln when the true strain range from 0.2 to the true strain corresponding to the maximum stress. The value of strain rate hardening item (m) can be obtained from the slope of the line in the ln − lṅ when the true strain is about 0.05. The value of softening item (s) can be obtained from the slope of the line in the ln − ln when the true strain corresponding to maximum stress and half of the maximum stress.

Phenomenological model including softening
The Fields-Backofen (FB) model is widely used and it can well express the work-hardening phenomenon by the strain hardening exponent (n) and the strain rate sensitivity exponent (m), which are the important parameters influencing the homogeneous deformation degree of metals during the hot forming [24]. But there are plenty of studies show dynamic recovery, recrystallization, globularisation softening, phase transformation and adiabatic heating occur simultaneously during hot deformation, which leads to FB model is inaccurate to describe the hot deformation behaviour of Ti-6Al-4 V titanium alloy [25]. Using the scheme of n, m and s in the above proposed hardening model, a previously determined softening item s is introduced to modify the FB model to describe the softening behaviours of titanium alloys under hot forming conditions. Considering temperature, deformation history and strain rate are the main factors that cause the softening of titanium alloy, therefore, the softening exponent (s) is a power function of temperature, strain and strain rate. The governing equation of new model is given as follows: where, is the flow stress; K is the strength coefficient; is the true strain; n is the strain hardening exponent.
Here, ̇ is the strain rate, ̇ * is the reference strain rate, set to 0.0001 s −1 . According to the literature, the strain hardening exponent ( n ), strain rate hardening exponent ( m ) and strength coefficient ( K ) in the Fields-Bachofen equation are functions of temperature ( T ) and strain rate ( ̇ ), which can be expressed by formula: where, a 1 , a 2 ,a 3 , b 1 , b 2 , c 1 , c 2 , c 3 , d 1 , d 2 , d 3 , e 1 , e 2 , e 3 , f 1 and f 2 are constants.

Prediction of uniform deformation model
In this work, the strain before localized necking point was used to describe the uniform strain for hot forming processes as the formulation from in Eq. (11). In order to calculate Eq. (11), the strain hardening exponent (n), strain rate sensitivity exponent (m) and softening exponent (s) is calculated by Eq. (12), Eq. (13) and Eq. (14), the results are presented in Table 2.
The experimental determined uniform strain using d ∕d = ∕2 and the predicted uniform strain using the Hill criterion (Eq. (9)), Hart criterion (Eq. (10)) and new criterion (Eq. (11)) are plotted in Fig. 9. The figure shows comparisons between the predicted and measured strain corresponding to localized necking moment under different forming temperatures (X-axis) and strain rates (right Y-axis). Left Y-axis is the errors of predicted and measured strain, defined as (E i − P i )∕E i , the smaller the errors is, the darker the colour is. It is clear the error of the Hill model and Hart model is almost over 200%, which is useless for reference. For further demonstrating the accuracy of the constitutive model, the absolute average relative error AARE has been used as an unbiased statistical parameter to evaluate the deviation of the predicted values [26,27]. This error is expressed as: where, E i is the experimental data, P i is the predicted date calculated using the obtained model and N is the number of data employed in this present investigation.
The calculated value of absolute average relative error AARE was 20.09% of new model, although there is a certain error between the measured and calculated value, the values may still provide useful straightforward guides for industries. The predicted uniform strain kept linear relation with the experimental determined uniform strain as show in Fig. 9d. Most of the data points lie close to the best regression line, and the correlation coefficient R for the predicted uniform strain model is 0.868. The closer R is to 1, the more accurate the prediction uniform strain is. Furthermore, the experimental error is believed to be caused by the use of phenomenological constitutive model. On one hand, material flow behaviour during hot forming process is often complex, but phenomenological constitutive model failed to consider physical mechanism of Ti-6Al-4 V sheet during high temperature deformation. On the other hand, in order to easily calibrate phenomenological constitutive model, the number of material constants is reduced, which leads to a larger error when used in a wide deformation range (temperature 973 to 1123 K, strain rate 0.01 to 1 s −1 ).
The strain and strain rate dual hardening of materials during thermal-mechanical process becomes a subject to be increasing noticed, especially to industrial applications. In future studies, the strain hardening, strain rate hardening and softening of materials during high temperature deformation will be quantitatively analysed on the microstructure point of view.

Quantification of hardening components
The relation between hardening exponents (n and m) and uniform strain at different temperatures based on the criterion of Hill and Hart is shown in Fig. 10. From the resultant the picture, the experimental determined uniform strain (the strain when d ∕d = ∕2 ) is considerably lower than predictions which takes hardening into account only. This is due to the softening phenomena play an important role during elevated temperature plastic deformation of Ti-6Al-4 V sheet, the rapid elimination of the accumulated dislocation density is caused by dislocation recovery and recrystallization, which leads to the uniform strain decreased with the increase of temperature and the minimum of uniform strain is 0.036 at 1073 K. Subsequently, the uniform strain increased to 0.042 at 1123 K due to the grain growth hardening [28]. At the same time, it is not difficult to find that strain hardening and strain rate hardening are useful tools to improve the deformation uniformity. In other words, the main measures of increase uniform strain are increase hardening exponents (n and m) to the most extent during the high temperature plastic deformation.
To evaluate the effect of temperature to the uniform deformation and deformation mechanisms under a range of temperatures (973-1123 K) for the strain rates of 0.1 s −1 , considering strain hardening individually (grey-shaded bars), the predicted uniform strain based on Hill criterion was decreased from 0.274 to 0.111 with the increasing temperature form 973 to 1073 K. The major factor that leads to the strain hardening behaviour is the continuous accumulation of the dislocations which may tangle and pining each other. However, the dislocations annihilated caused by dynamic recovery and recrystallization, which may decrease the strain hardening exponent (n) and decrease the uniform strain, which affects deformation behaviour of Ti-6Al-4 V sheet. But the temperature rises to 1123 K, the grain rotation slowed down the annihilation of dislocations, which leads to strain hardening exponent (n) and uniform strain increased. The thermally activated escape of pinned dislocations is the main mechanism of strain rate hardening, therefore the strain rate hardening exponent (m) raised with temperature and show good temperature dependence, as shown in Table 2. According to the formula (10), it is concluded that the uniform strain increased with the increases of strain rate hardening exponent (m). The uniform strain is only increased by 15% at 973 K while the uniform strain is increased by 33% at 1123 K. The change law of softening exponent (s) is exactly the opposite of the change law of strain hardening exponent (n), this has confirmed the strain hardening by dislocation density accumulation and the softening behaviour by recovery and recrystallization jointly influence the deformation behaviour of Ti-6Al-4 V sheet under the high temperature deformation conditions. Obviously, the softening is detrimental to the uniform deformation, the uniform strain is reduced about eighty percent resulting from the recovery and recrystallization that soften the flow stress.

Phenomenological model prediction and hot flow behaviour
In order to verify the derived phenomenological model including softening, the determined constants in Table 3 were used to make the model predictions of flow stress curve. Figure 11 shows the comparison between predicted and measured flow stress curves of Ti-6Al-4 V sheet under three different strain rates and forming temperatures. It can be easily found that FB model is inaccurate to describe the softening behaviour of Ti-6Al-4 V sheet during hot deformation, In contrast is the proposed new constitutive equations give an accurate and precise estimate of the flow stress for Ti-6Al-4 V sheet. As is evident, most of the data points predicted by new model near to the beat regression line as show in Fig. 11b. The correlation coefficient R for the predicted phenomenological model is 0.997 and the calculated value of absolute average relative error AARE was 3.75% much better than the FB model (R = 0.980%, AARE = 14.07%), which indicates that the softening exponent (s) is introduced into the FB equation can give an accurate estimate of the flow stress and plastic deformation behaviour for Ti-6Al-4 V sheet at elevated-temperature and can be used to numerically analyse the hot working process of this material.
The results were indicative that the temperature has an important impact on true strain. It was found that the true strain of Ti-6Al-4 V sheet is quite low when the temperature below 973 K. However, for temperatures greater than 1023 K, the true strain increased dramatically from 0.385 to 0.49 at 1123 K as shown in Fig. 11a. This is mainly due to atomic vibrating energy increase with the increase of temperature, promote coordinated deformation between α phase and β phase and restrains the formed of micro-voids. Beyond that, temperature also affects the value of hardening exponents (n and m). The result shows in Table 2, as the temperature rises the strain hardening exponent (n) gradually decrease to the minimum of 0.055 at 1073 K, and then increased with the increase of temperature. The dynamic recovery caused by sliding or climbing of dislocation below 1073 K and the rotational dynamic recrystallization at 1123 K is responsible for the phenomenon [14,29]. The strain rate hardening exponent (m) increased monotonously with temperature increasing might be interpreted as thermally activated escape of pinned dislocations. This view was also used to explain the increase of ductility at elevated temperature by Kopec et al. [30]. It should be mentioned that the significant strain hardening behaviour can be observed below 973 K, leading to an ultimate tensile strength of 435.46 MPa. At 1023 K and higher temperature, the flow stress curve become flat due to the occurrence of materials softening, indicating that the deformation temperature play an important role to the deformation mechanism. The migration of the grain boundaries may cause the grain sizes reduced, which may soften the flow stress under the hightemperature deformation conditions, as suggested in Refs [31,32]. Additionally, the smaller grain size may also result in the softening of flow stresses [33]. Hence, the dynamic recrystallization may compensate a small fraction of material hardening from the accumulation of the dislocation, leading to a relatively 'flat' stress-strain curve. Figure 11 presents the effect of strain rate on the hot flow behaviour also. In general, the ductility of the Ti-6Al-4 V sheet is increased with decreasing strain rate. For a given temperature, i.e. temperature of 1023 K, the strain to failure was increased from 0.25 to 0.41 with the strain rate decreases from 1 to 0.01 s −1 , and this trend can be also found in the temperature of 1123 K. Therefore, the view that the influence of strain rate on ductility of the Ti-6Al-4 V sheet become negligible when the strain rate lower than 0.1 s −1 . For the flow stress level, maximum flow stress decreases with the decrease of strain rate at same temperatures (e.g. 1023 and 1123 K), and it was found that the softening phenomenon is enhanced with decreasing strain rate. This phenomenon is related to the dislocation mobility and atom diffusion at lower strain rate resulting in strongly strain rate sensitive. Through the change of the strain hardening exponent (n) list in Table 2, it is not difficult to find this view is correct, the strain hardening exponent (n) decreases with the decrease of strain rate for the two groups. It is worth noting that the strain hardening exponent (n = 0.183) at 1123 K (strain rate of 0.01 s −1 ) is a special case. It is the largest of all the strain hardening exponents, it can be attributed to grain growth hardening. Ghosh and Hamilion [34] performed systematic experiments and found that grain growth plays an important role in the hardening of high temperature deformation mechanisms which leads to flow stress increased with increasing strain at constant strain rates.

Conclusions
The work described in this paper proposed efficient models of quantifying hardening and constitutive behaviour of hot forming titanium alloys. The quantification of strain, strain rate hardening and softening contributes to understanding of deformation uniformity under hot forming conditions and further facilities the determination of process parameters for engineering applications. The following conclusions can be drawn: (1) A new quantification model of hardening was proposed to predict uniform strain considering strain hardening, strain rate hardening and softening for the first time.
An acceptable prediction was achieved estimation as the correlation coefficient R is 0.868 and the absolute average relative error AARE was 20.09% for certain conditions. (2) Precise hot uniaxial tensile behaviours of Ti-6Al-4 V were characterized using the established volumeconsistency based correction method, which enables the issue of effect of non-uniform temperature within specimen of Gleeble testing to be solved. (3) A modified phenomenological model was formulated to predict the hot flow behaviour of Ti-6Al-4 V sheet, with typical softening feature being taken into account. Good agreements were found between experimental results and model predictions. The absolute average relative error AARE of novel phenomenological model was 3.75% and the correlation coefficient R is 0.997.