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S0264-1275(16)30741-9 doi: 10.1016/j.matdes.2016.05.118 JMADE 1866

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Please cite this article as: Nima Haghdadi, David Martin, Peter Hodgson, Physicallybased constitutive modelling of hot deformation behavior in a LDX 2101 duplex stainless steel, (2016), doi: 10.1016/j.matdes.2016.05.118

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ACCEPTED MANUSCRIPT Physically-based constitutive modelling of hot deformation behavior in a LDX 2101 duplex stainless steel

Institute for Frontier Materials, Deakin University, Geelong, VIC 3216, Australia 2 SwereaKIMAB, Box 7047, Kista, SE-16407, Sweden

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Nima Haghdadi1, David Martin2, Peter Hodgson1

Dedicated to the late Dr. Jan-Olof Andersson.

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Abstract

A detailed understanding of the hot deformation and work hardening behavior of

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LDX 2101 dual phase steel has been obtained through a wide range of hot compression tests with strain rates from 0.01 to 50 s-1 and temperatures from 900 to 1250°C. In most of the

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cases, the material showed typical dynamic recrystallization (DRX) behavior i.e., a peak

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followed by a gradual decrease to a steady state stress. The work hardening rate showed a

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two stage behavior i.e., a transient sharp drop at low stress values followed by a gradual decrease at higher stresses. Using the work hardening rate behavior at the latter stage, the saturation stress was calculated for different hot working conditions. Regression methods

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were used to develop a hyperbolic-sine equation linking the saturated stress to the deformation conditions. A physically-based Estrin-Mecking (EM) constitutive equation was then employed to model the flow behavior in the work hardening (WH)-dynamic recovery (DRV) regime. Finally, the Avrami equation to describe the evolution of the softening fraction was coupled to the EM model to extend the model to the dynamic recrystallization region. The results show that the model which is based on the stress-strain and work hardening behavior accurately predicts the flow behavior of this microstructurally complex steel.

Corresponding Author. E-mail address: [email protected]

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ACCEPTED MANUSCRIPT Keywords: Duplex stainless steel; Dynamic recrystallization; Constitutive equation; EstrinMecking equation; Avrami equation

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1. Introduction Duplex stainless steels with a microstructure composed of austenite and ferrite offer

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an attractive combination of strength and toughness as well as high corrosion resistance. This is why a significant metallurgical research over the past few years has been devoted to design

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and study of physical and mechanical properties of this grade of steels. LDX 2101 (EN 1.4162, UNS S32101) is a relatively new duplex stainless steel with mechanical

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characteristics and corrosion resistance comparable to standard single and dual phase stainless steels [1]. A low alloying element content, particularly Ni, makes this grade more

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economical over other common duplex stainless steels. Also the low alloying element content

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makes this grade less prone to the formation of undesirable precipitates and intermetallics [2].

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Accordingly, LDX 2101 would be a cost-effective potential candidate to be used in diversified applications such as structural members and reinforcement bars as well as

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chemical process vessels, piping and heat exchangers. Hot working is an important step in producing dual phase steel sheet and plate. In comparison to single phase steels, the deformation of dual phase steels is much more complicated where due to the different co-existing crystal structures and stacking fault energy values; ferrite and austenite show very different responses to external loading at high temperatures [3,4]. This necessitates a detailed study of hot deformation of these steels. Advanced analysis of the hot working behaviour relies on knowledge of the flow of the material under different thermomechanical conditions. This is typically performed through constitutive analysis where a series of mathematical equations describe the instantaneous response of the material to loading as a function of the process variables and material

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ACCEPTED MANUSCRIPT structure [5,6]. The development of accurate but simple constitutive equations for deformation behavior of materials has driven a significant amount of research over the past

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few decades.

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In the literature there is a wide range of constitutive equations employed for different

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materials. In general, however, these equations could be classified into three main groups i.e., empirical, semi-empirical and physically-based models [5]. The two former categories are

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based on regression methods and are often straightforward [7-9]. Nevertheless, they lack any physical meaning. On the other hand, physically-based models offer an equation which takes

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the structure of the material and micromechanisms of deformation into account, although they are more complicated [10]. Kocks [11] and later Estrin and Mecking [12] have proposed

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a physical model in which the material resistance to flow was considered as a transient

where

in which

is the flow stress,

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as

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process from an initial state to a stationary final state. In a simple way this could be described

parameter representing the physical state of the material,

is the structural

is strain rate and T is the

temperature. The structural parameter was then related to the dislocation density as the

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internal variable. Incorporation of dislocation density is important as it is the dislocation evolution and movement that causes both the deformation and strain hardening [13]. The related parameters for considering the dislocation evolution in the model are obtained through analysis of the work hardening behavior of the material. Based on the Estrin-Mecking (EM) analysis, in the absence of dynamic recrystallization (DRX), the flow characteristics of the materials is dictated by the hardening effect of dislocation generation and tangles and softening effects of recovery during which dislocations rearrange and annihilate [14]. This will then result in a typical stress-strain behavior with convergence to a steady state at a saturation stress without any stress decrease.

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ACCEPTED MANUSCRIPT Many metals, however, dynamically recrystallize during hot deformation. The most widely adopted approach to incorporate the softening induced by DRX in the constitutive equation is

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to employ the Avrami equation [15]. Based on this approach, the softening induced by DRX

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is calculated and incorporated to the EM saturation stress values for any strain value beyond

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εc (the critical strain for DRX).

Taking all the above into account, the objective of this work is to establish an

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appropriate constitutive relationship between the flow stress and deformation parameters to predict the hot deformation behavior of LDX 2101 dual phase steel. To achieve this, a wide

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range of hot compression tests were carried out. Using stress strain and the corresponding work hardening curves, the relationship between structural and deformation parameters and

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characteristic stress and strain values were determined. Finally the modified EM constitutive

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equation coupled with the Avrami equation was used to model the flow stress of the duplex

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stainless steel.

2. Experimental procedure

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Production LDX 2101 steel with the chemical composition of 21.5Cr, 1.5Ni, 0.03C, 5Mn, 0.22N, 0.3Mo, 0.7Si, 0.35Cu, and remainder Fe (wt.%) was received as hot rolled transfer bar, approximately 26mm thick. Cylindrical compression samples were prepared from this material in size of Φ10 × H15. The hot uniaxial compression tests were conducted in the temperature range of 900–1250 °C under the strain rates of 0.01, 0.1, 1, 10 and 50 s−1. Compression tests were done using a Gleeble® 3800 Hydrawedge machine. Before straining, the specimens were heated at 5°C/s and held at the deformation temperature for 2 min to equalize the temperature throughout the samples. The hot compression tests were then carried out up to a strain of about 0.7. 3. Results and discussion 4

ACCEPTED MANUSCRIPT 3.1. Hot compression behavior The true stress-strain curves obtained during hot compression of the experimental

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material at different temperatures under various strain rates are given in Fig. 1 (a-e). The

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curves show the classical DRX-accompanied flow shape i.e., initial work hardening and a

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peak followed by softening beyond the peak. More detailed observation shows that a yield point like phenomenon is observed at the working temperature of 1250 °C (see Fig. 1).

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Dehghan-Manshadi et al. [16], and Duprez et al. [17] also observed such a behavior in the stress-strain curves of 2205 duplex steel but over a wider working temperature range. This

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has been proposed to be due to strain partitioning between austenite and ferrite. However, this may also be the result of dynamic strain ageing (DSA) [18].

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In duplex stainless steels, strain is mostly accommodated by ferrite. This is because

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ferrite is of a higher stacking fault energy, so arrangement and annihilation of dislocations

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would occur easier in ferrite. As deformation proceeds, there is a tendency to transfer the load towards austenite. However, as austenite/ferrite interphase becomes incoherent with deformation, the slip transfer from ferrite to austenite becomes difficult resulting in the most

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of strain remaining in ferrite. For duplex stainless steels, there is a controversy over the main DRX mechanism being active for austenite and ferrite under different hot working conditions. In case of austenite, typical DDRX has been reported to take place. At sufficiently high temperatures, however, Cizek [3] has proposed that subgrain coalescence would also contribute to austenite dynamic softening. Ferrite softening mechanism has been also shown to be highly influenced by the Z parameter. Castan et al [19], have suggested that ferrite goes through DDRX at high Z values while it softens by CDRX under low Z values. This has been supported in a recent work by Cizek [3].

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ACCEPTED MANUSCRIPT In general, an increase in the flow stress level and the peak stress is observed when increasing the strain rate and/or decreasing the working temperature. This has been quantified

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in Fig. 2 where the variation of peak stress with temperature and strain rate is shown. The

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effect of deformation rate on the flow stress level is reasoned considering the higher amount

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of tangled dislocation structures at higher strain rates. As is known, these tangles hinder dislocation movement and hence increase the resistance of the material to deformation. The effect of temperature is rationalized as follows. It is generally accepted that the volume

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fraction of ferrite increases with temperature in duplex stainless steels. As the ferrite phase is

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the softer phase, this may result in overall softening of the material [18]. In addition, the stress for dislocation breakaway from the pinning points is decreased at elevated temperatures

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[20]. In the post work hardening flow behavior regime, where the restoration takes place, this

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can be explained considering the thermally activated nature of the restoration phenomena. Fig.3 shows the effect of the deformation parameters (strain rate and temperature) on

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the peak strain value. The peak strain increases at lower deformation temperatures and higher strain rates. As a rule, the peak occurs when DRX softening overcomes the work hardening

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effect. As a thermally activated phenomenon, DRX is delayed as the temperature decreases, which in turn shifts the peak strain to higher strains. The diffusional nature of DRX also results in a delay of the softening at higher strain rates. This is because less time is available for restoration to proceed at high deformation rates. 3.2. Work hardening behavior Study of the work hardening behavior of a material can provide great insights into the deformation and restoration characteristics of the material. This will be discussed in this section based on the Kocks-Mecking approach [21]. In this regard the evolution of the strain hardening rate ( =d /d ) with stress for the examined material under different strain rates

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ACCEPTED MANUSCRIPT and temperatures is given in Fig.4. As expected, the work hardening rate decreases with decreasing the strain rate and increasing the temperature. This is because the rate of work

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hardening is dictated by the competition between storage and annihilation (rearrangement) of

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dislocations, and dynamic recovery is more pronounced at higher temperatures and lower

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strain rates [22].

In all the curves, the strain hardening rate decreases with increasing stress as

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deformation continues. In detail, the hardening behaviour can be characterized by a rapid drop at low stresses (transient stage) followed by a gradual linear decrease at high stresses.

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The strain hardening rate then decreases to zero (for stress values close to the saturation stress) and then falls below zero indicating the occurrence of dynamic recrystallization. In a

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modified Kocks-Mecking model [11], the dislocation density evolution is considered as the

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main factor governing the deformation behavior in the linear stage of the work hardening rate

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curves. In this stage, the instantaneous rate of work hardening can be expressed as in which

is the initial work hardening rate and

strains. In the current work the

is the saturation stress at high

and saturation stress values were extrapolated for all the

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deformation conditions through fitting linear equations of the latter stage of work hardening. A schematic of the saturation stress in stress-strain curves and work hardening curves is shown in Fig.5. 3.3. Hyperbolic-sine equation It has been proposed by Sellars and Tegart [23] that a hyperbolic sine-type equation can be employed to correlate the saturation stress to the deformation conditions over a very wide range of stresses: (1)

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ACCEPTED MANUSCRIPT where σs is the saturation stress (MPa),

is the strain rate (s−1), Q is the apparent

activation energy for deformation (J mol−1), R is the universal gas constant (8.314 J mol-1K-1),

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and A (s−1), α (MPa−1) and n are material constants which need to be obtained experimentally.

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For different levels of strain, more simple equations can be used to describe the flow

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curves. In particular the following equations have been shown to model the flow behavior for

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low and high stress levels, respectively:

(2) (3)

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where B and C are material constants. It has been reported that the alpha value in equation 1 is determined by dividing β by n1. In order to calculate the latter parameters the

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(4)

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temperature as follows:

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logarithm of the exponential law and power law constitutive equations were taken at constant

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(5)

The relationship between saturation stress and strain rate at different temperatures is given in Fig.6. Through linear data fitting and making averages, the β and n1 values were calculated to be 0.069 and 6.73 which yields an α value of 0.010. In order to calculate the n, A and Q values, it is necessary to take the logarithm from both sides of Eq. (1) which yields: (6)

As seen in Fig.7, plotting ln [sinh(ασ)] versus ln ( ) and taking the average slope of the lines would yield n. Finally rearranging equation 6 gives the following equation: 8

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Through averaging the slopes of ln [sinh (ασ)] vs. 1/T under different strain rates in

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Fig.7, the apparent activation energy of the deformation is calculated. Finally, knowing the values of n and Q, ln A is easily found from the interception of ln[sinh(ασ)] vs. ln . The

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obtained n, Q and α values for the experimental steel are given in Table 1 and have been compared to the values obtained by other researchers during constitutive analysis of different

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grades of duplex stainless steels. As is seen, the values obtained for the steel in this study falls between those calculated for other dual phase steels. However, some discrepancies are

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evident even for quite similar alloys. The source of these differences can be due to errors raised when using linear regression methods. The complicated nature of phase transformation

Considering

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and precipitation in this kind of steels can also be another source for such differences. , the saturation stress under different thermomechanical

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conditions can be obtained through the following equation:

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(8)

Fig. 8 provides a comparison between the experimental and predicted saturated stress values. As is seen a very good agreement has been obtained which shows the applicability of the latter equation in estimating the saturated flow stress under different thermomechanical conditions. The predictability of the equation was examined using average absolute relative error parameter (AARE=

), where E is the experimental finding, P is the

predicted value, and N is the number of data which were employed in the fitting. The AARE was found to be 5.09 %, which shows an accurate estimation of the flow stress. 3.4. Estrin-Mecking equation 9

ACCEPTED MANUSCRIPT Based on the Kocks-Mecking model, the apparent hardening depends on the dislocation density which is the result of interplay between storage and annihilation of dislocations as

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follows [11]: (9) (10)

The combination of these equations results in a linear relation between work hardening rate

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and the flow stress after the transient step, as follows:

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where

(11)

2 which is strain rate independent but temperature dependent as the G . Here,

is a numerical

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(shear modulus) is temperature dependent, and

constant, b is the magnitude of the Burgers vector and coefficients k1 and k2 represent the

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dislocation storage and annihilation rate, respectively. By taking the assumption that the storage rate is constant, Estrin and Mecking [12] modified the KM model and proposed that:

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(12)

Which then yields to the following equation: (13) After integration, the flow stress in work hardening and dynamic recovery regime can be given by (14) And the saturation stress is given by

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ACCEPTED MANUSCRIPT (15)

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In order to predict the flow stress based on the modified EM model, the B constant should be

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calculated. Two approaches can be used to calculate B. One way is through using equation versus

can yield B for

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(14) where linear regression of the variation of

different thermomechanical conditions. B value also can be calculated through analysis of the curves where linear regression of variation of

versus

gives B. In the current work,

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the first approach was utilized and an example curve used for extrapolating B is shown in

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Fig11. As expected the B value was almost independent of strain rate. The dependency of the B on temperature was also very low to the extent that it can be ignored

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Calculating the B value (almost=7 for this steel) through Fig.9, the flow curves of the

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material can be predicted through EM model up to the peak strain where the most important

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factors controlling the flow are work hardening and dynamic recovery. In order to extend the model to the softening region, the dynamic softening of the material needs to be incorporated into the model.

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3.5. Incorporation of the Avrami equation In order to compensate for the softening effect of dynamic recrystallization in the constitutive equation, the volume fraction of DRX must be known for each strain after the critical strain for recrystallization (or after the peak strain in a more straightforward manner). This can be calculated for every strain using the following expression [32]. (16)

where

is the flow stress predicted by the above Estrin-Mecking model and

is the

actual steady state stress. Based on the Avrami equation, the DRX volume fraction can be estimated using the expression below. 11

ACCEPTED MANUSCRIPT (17) Variations of

versus

can then be used to calculate the a and b

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values. In the current work these constants were calculated to be 10 and 1.5, respectively.

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Finally, the combined Estrin-Mecking and Avrami equation used to model the flow behavior

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of the material over the entire strain range considering different regions of work hardening,

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dynamic recovery and dynamic recrystallization can be summarized as:

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Note that in this work, similar to previous works [33], the critical strain (

(18) (19) is replaced by

the peak strain ( ) due to the difficulty and uncertainties in accurately determining this

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quantity for some of the deformation conditions. It is to be noted that the model can be more

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generalized by making all constants a function of ln (Z). Some of these correlations have

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been given in Fig 10. Finally, the predictability of the proposed coupled model is shown in Fig.11. As is seen, a good prediction is obtained in both before peak (EM model) and after peak (EM+Avrami). Some deviation from the experimental results is observed at high stress

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values which may show that at high stresses, the work hardening curve cannot well indicate the flow behavior characteristics. By using the developed model, and employing the general correlations between Z parameter and characteristics parameters in the model, a good estimate of the flow behavior for the examined material can be achieved knowing just the temperature and strain rate at which the deformation is imposed to the material. 4. Conclusion A detailed study of the hot compression and work hardening behavior of duplex stainless steel LDX 2101 was carried out over a wide temperature and strain rate range. The stressstrain curves showed typical dynamic recrystallization (DRX) behavior i.e., a peak followed

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ACCEPTED MANUSCRIPT by a gradual decrease to a steady state stress. The work hardening rate evolution consisted of two stages: a transient sharp drop at low stress values followed by a gradual decrease at

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higher stresses. It was shown that the peak stress and strain as well as the rate of work

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hardening are coupled to the strain rate and temperature. The peak stress, peak strain and

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work hardening rate increased with an increase in strain rate and decrease in temperature. The hot compression behavior of the studied material was then modeled using a modified EstrinMecking approach. The related structural and phenomenological parameters were extracted

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using the hot compression and work hardening data. It was shown that the modified EM

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model can predict the flow behavior up to the peak stress. In order to extend the model to dynamic recrystallization region, a softening term was introduced to the model. This was

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done through coupling the Avrami equation to the EM model. Finally, through comparing the

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experimental results with the predictions of the model, the accuracy of the proposed model

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was confirmed.

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Acknowledgements

This paper is dedicated to the memory of Dr. Jan-Olof Andersson, who made a significant contribution to the computational thermodynamics and physical metallurgy of stainless steels and the metals industry in Sweden, and who instigated and supported the research from which this paper stems. Parts of this work were performed and funded under the auspices of the Swedish National Action for Metallic Materials funded “Profroll” project. The authors would also like to acknowledge the support of Outokumpu Stainless AB, in particular Cecilia Lille, for granting permission to publish this work.

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Figure Captions

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Fig.1. The true stress–true strain curves of the studied steel obtained by hot compression tests at strain rates of (a) 0.01s−1; (b) 0. 1 s−1; (c) 1 s−1; (d) 10 s−1 and (e) 50 s−1. Fig.2. The variation of peak stress vs. deformation temperature under different strain rates. Fig.3. The variation of peak strain vs. deformation temperature under different strain rates. Fig.4. The work hardening rate behavior of the examined steel under different temperatures and strain rates. Fig.5. The Schematic defining and showing how to calculate saturation stress σs and θ0. Fig.6. Evaluating the value of (a) n1 by plotting ln σs vs. and (b) β by plotting σs vs.

Table caption

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Fig.7. Evaluating the value of (a) n by plotting ln[sin h(ασ)]ln[sin h(ασ)] vs. and (b) Q by plotting ln[sin h (ασ)] vs. 1/T. Fig. 8. The correlation between the experimental and predicted saturation flow stresses using the sin-hyperbolic equation. Fig.9. The variations of vs. strain to calculate B value. Fig.10. The variations of (a) peak stress and (b) peak strain as a function of Zener–Holloman parameter. Fig. 11. The comparisons between predicted (colour lines) and measured flow stress curves (black lines) of experimental steel at temperatures of (a) 900 °C and (b) 1100 °C.

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Table1. The calculated n, Q and α parameter for some duplex stainless steels.

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(a)

900 °C 900 °C

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1000 °C

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1000 °C 1100 °C 1200 °C

(c)

1100 °C 1200 °C 1250 °C

900 °C

900 °C 1000 °C 1100 °C

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1100 °C

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(d)

1200 °C 1250 °C

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1200 °C 1250 °C

(e)

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900 °C 1000 °C

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1100 °C

Fig.1. The true stress–true strain curves of the studied steel obtained by hot compression tests at strain rates of (a) 0.01s−1; (b) 0. 1 s−1; (c) 1 s−1; (d) 10 s−1 and (e) 50 s−1.

17

NU

SC R

IP

T

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AC

CE P

TE

D

MA

Fig.2. The variation of peak stress vs. deformation temperature under different strain rates.

18

NU

SC R

IP

T

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AC

CE P

TE

D

MA

Fig.3. The variation of peak strain vs. deformation temperature under different strain rates.

19

NU

SC R

IP

T

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Fig.4. The work hardening rate behavior of the examined steel under different temperatures

AC

CE P

TE

D

MA

and strain rates.

20

NU

SC R

IP

T

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AC

CE P

TE

D

MA

Fig.5. The Schematic defining and showing how to calculate saturation stress σs and θ0.

21

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(b)

NU

SC R

IP

T

(a)

AC

CE P

TE

D

MA

Fig.6. Evaluating the value of (a) n1 by plotting ln σs vs.

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and (b) β by plotting σs vs.

.

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(a)

SC R

IP

T

(b)

NU

Fig.7. Evaluating the value of (a) n by plotting ln[sin h(ασ)]ln[sin h(ασ)] vs.

AC

CE P

TE

D

MA

plotting ln[sin h (ασ)] vs. 1/T.

23

and (b) Q by

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MA

NU

SC R

IP

T

AARE= 5.09%

D

Fig. 8. The correlation between the experimental and predicted saturation flow stresses

AC

CE P

TE

using the sin-hyperbolic equation.

24

AC

CE P

TE

D

Fig.9. The variations of

MA

NU

SC R

IP

T

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25

vs. strain to calculate B value.

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(a)

SC R

IP

T

(b)

NU

Fig.10. The variations of (a) peak stress and (b) peak strain as a function of Zener–Holloman

AC

CE P

TE

D

MA

parameter.

26

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(a)

(b) 10 s-1

10 s-1 1s

-1

SC R

IP

T

0.1 s-1

1 s-1 0.1 s-1

NU

Fig. 11. The comparisons between predicted (colour lines) and measured flow stress curves (black

AC

CE P

TE

D

MA

lines) of experimental steel at temperatures of (a) 900 °C and (b) 1100 °C.

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T=900 °C 10 s-1

T

1 s-1

NU

SC R

IP

0.1 s-1

AC

CE P

TE

D

MA

Graphical abstract

28

ACCEPTED MANUSCRIPT Table1. The calculated n, Q and α parameter for some duplex stainless steels.

Q

Mode of

n

T

Reference

α

(kJ/mol) (MPa−1)

IP

deformation

526

0.010

6.9

569

0.009

4.0

438

0.007

torsion

4

474

0.012

torsion

4.2

450

0.0139

Plane strain

4.02

578

0.0142

torsion

5.63

536

0.012

torsion

3.8

407

0.012

uniaxial

6:62

460

0:012

5.1

479

0.012

4.4-4.7

482-512

0.011-

uniaxial

4.85

SC R

2101 (This work)

compression uniaxial

2205 (Cabrera et al. [24])

NU

compression uniaxial

2507 (Cabrera et al. [24])

MA

compression

D

2205 (Spigarelli et al. [25])

CE P

TE

23Cr, 4.8Mn, 0.22Mo (Iza-Mendia et al. [26])

compression

2304 (Evangelista et al. [27])

AC

2205 (Evangelista et al. [27]) 2205 (Yang et al. [28])

compression uniaxial

2205 (Momeni and Dehghani [29]) compression uniaxial

2205 (Momeni et al. [30]) compression tension

2205 (Faccoli et al. [31])

29

0.013 4.9

430

0.014

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Highlights

Saturated stresses were measured using hyperbolic-sine equation

T

Hot deformation and work hardening behavior of LDX 2101 was studied

IP

Work hardening-dynamic recovery regime was modelled by Estrin-Mecking equation

AC

CE P

TE

D

MA

NU

SC R

Avrami equation was coupled to the Estrin-Mecking to model the DRX region

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