Analysis of flow behaviour and strain partitioning mechanism of bimetal composite under hot tensile conditions

International Journal of Mechanical Sciences 000 (2020) 105317

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Analysis of flow behaviour and strain partitioning mechanism of bimetal composite under hot tensile conditions Zhou Li a, Jingwei Zhao a, Fanghui Jia a, Yao Lu a, Qingfeng Zhang b, Sihai Jiao b, Zhengyi Jiang a,∗ a b

School of Mechanical, Materials, Mechatronic and Biomedical Engineering, University of Wollongong, Wollongong, NSW 2522, Australia Baosteel Research Institute (R&D Centre), Baoshan Iron & Steel Co., Ltd., Shanghai 200431, China

a r t i c l e

i n f o

Keywords: Bimetal composite Hot flow behaviour Strain partitioning Constitutive model Mixture law

a b s t r a c t The flow behaviour and strain partitioning of 2205 duplex ferritic-austenitic stainless steel/AH36 low carbon steel bimetal composite (2205/AH36 BC) were investigated at elevated temperatures in this study. A physicallybased constitutive model was established to describe the flow behaviour of bimetal composite based on the stress–strain relationships obtained from hot tensile tests, which were performed on a Gleeble 3500 thermalmechanical test simulator over the temperature range of 950–1250 °C and strain rate range of 0.01–1 s−1 . The stress and strain partitioning of bimetal composite was analysed to develop a mixture law of flow stress under those hot working conditions, due to the different strain contributions of each material on the total flow behaviour of bimetal composite. The developed constitutive model and the mixture law considering strain partitioning were both adopted to predict the stress–strain curves and used in finite element (FE) simulation model to calculate the peak loads of 2205/AH36 BC at 1000 °C. It is found that the softening mechanisms of 2205/AH36 BC changes depending on the externally imposed working temperatures and strain rates. The contribution of AH36 carbon steel layer on total stress is relatively more than that of 2205 stainless steel layer at high temperatures, as a result of the occurrence of stress partitioning. The proposed constitutive model, instead of the mixture law, is recommended to be used in the FE simulation of practical hot working of bimetal composite due to its high accuracy.

1. Introduction Bimetal composite made by bonding such as stainless steel to a carbon or alloy steel can offer the benefits of both bonded constituent metals, including the mechanical properties of the carbon steel and the corrosion resistance of the stainless steel. Consequently, bimetal composite has been used in a variety of industrial fields including shipbuilding, refineries, chemical, oil and gas production, as well as power plants and mining equipment [1–4]. However, the plate thickness of such bonded composite is generally not small, resulting in its hard workability in practice. In order to solve the processing problem, the hot processing methods of metals, such as hot forming, spinning, and forging, are extensively employed to fabricate the final products with required geometry and properties. An in-depth understanding of hot deformation behaviour of materials is critical to the optimisation of industrial hot working processes, thereby obtaining the desired microstructure and optimum mechanical properties [5–7]. The hot deformation behaviour of metals is often complex due to the various deformation mechanisms including work hardening (WH), dynamic recovery (DRV) and dynamic re-

∗

crystallisation (DRX) at elevated temperatures [8–10]. Over the past decades, based on thermo-mechanical experiments, constitutive models have been widely adopted to describe the relationship between the hot flow behaviour and the processing parameters under hot working conditions [11]. Among them, phenomenological plastic constitutive models, including Johnson-cook (JC) model [12,13], Khan-Huang (KH) model [14], Fields-Backofen (FB) model [15] and Arrhenius equation [16,17], are the most widely used to express the dependence of flow behaviour on processing parameters. Phenomenological constitutive models, however, are lack of the physical background and just fit a convenient mathematical function. Therefore, many physically-based constitutive models, including Zerilli–Armstrong (ZA) model [18], as well as DRV and DRX models [19,20], which involve in thermodynamic theory, dislocation mechanisms and kinetics slips, have been presented by a number of researchers. These constitutive models were mostly developed based on the hot compression tests, but the relationship between flow stress and processing parameters for sheet/plate materials is usually generated by the tensile tests considering the effects of necking and damage evolution [21]. Cheng et al. [15] and Guo et al. [22] investigated the hot tensile flow behaviour of magnesium and titanium alloys, respectively. Li et al.

Corresponding author. E-mail address: [email protected] (Z. Jiang).

https://doi.org/10.1016/j.ijmecsci.2019.105317 Received 7 August 2019; Received in revised form 4 November 2019; Accepted 9 November 2019 Available online xxx 0020-7403/© 2019 Elsevier Ltd. All rights reserved.

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International Journal of Mechanical Sciences 000 (2020) 105317

Nomenclature 𝜎p 𝜎c 𝜎s 𝜎 ss 𝜎0 𝜎 necking 𝜎 𝜀 𝜀c 𝜀p 𝜀̇ 𝜌 𝜌0 U Ω M 𝛼0 𝜇 b Z ZAH36 Z𝛿 Q R T A 𝛼 𝜀t n XD k c 𝜎 total 𝜎 2205 𝜎 AH36 𝜎𝛾 𝜎𝛿 𝜀̇ 𝑡𝑜𝑡𝑎𝑙 𝜀̇ 2205 𝜀̇ 𝐴𝐻36 𝜀̇ 𝛾 𝜀̇ 𝛿 f2205 fAH36 f𝛾 f𝛿 Ω𝛾 Ω𝛿 𝜎 ptotal 𝜎 pAH36 𝜎 p2205 𝜎 p𝛾 𝜎 p𝛿 DRX DRV WH SFE PAGs

peak stress critical stress saturation stress steady state stress initial stress stress at necking point flow stress plastic strain critical strain peak strain strain rate dislocation density initial dislocation density work hardening coefficient softening coefficient Schmidt orientation factor proportional constant shear modulus burgers vector Zener-Hollomon parameter Zener-Hollomon parameter of AH36 Zener-Hollomon parameter of 𝛿-ferrite activation energy gas constant absolute temperature material constant material constant total strain material index volume fraction of DRX material constant material constant stress of bimetal composite stress of 2205 layer stress of AH36 layer flow stress of austenite flow stress of 𝛿-ferrite strain rate of bimetal composite strain rate of 2205 layer strain rate of AH36 layer strain rate of austenite Strain rate of 𝛿-ferrite volume fraction of 2205 layer volume fraction of AH36 layer volume fraction of austenite volume fraction of 𝛿-ferrite contribution coefficient of austenite contribution coefficient of 𝛿-ferrite peak stress of 2205/AH36 BC peak stress of AH36 peak stress of 2205 peak stress of austenite peak stress of 𝛿-ferrite dynamic recrystallisation dynamic recovery work hardening stacking fault energy prior austenite grains

[23] studied the flow softening and ductile damage of TC6 alloy using hot tensile tests and found the difference of flow curves between hot tensile and compressive tests.

Fig. 1. Schematic illustration of the bimetal composite consisting of 2205 layer with the thickness of 3 mm and AH36 layer with the thickness of 8 mm.

However, previous works were almost focused on the hot deformation behaviour or microstructural evolution of pure metal or single material. Less investigation has been performed so far on the flow behaviour of bimetal composite under hot tensile conditions. Moreover, the flow behaviour of bimetal composite at elevated temperatures may be different from that of its any constituent metal. As the result of the difference in strength between the carbon steel and stainless steel, for example, the different hot straining responses between them give a rise to the strain and stress partitioning, especially at the early stage of strain accumulation [24]. A possible consequence of this situation is that the flow behaviour of different internal zones is inconsistent with the total flow behaviour of bimetal composite. Several researchers have studied some work on the strain and stress partitioning of different phases in single metal, such as duplex stainless steel and two-phase titanium alloy [24–26], but no relative research has been reported for the bimetal composite to date. Therefore, this study aims to investigate the deformation behaviour of 2205 duplex ferritic-austenitic stainless steel/AH36 low carbon steel bimetal composite (2205/AH36 BC) over the temperature range of 950– 1250 °C and strain rate range of 0.01–1 s−1 . A physically-based constitutive model was established to predict the flow behaviour of 2205/AH36 BC during hot working, and the accuracy was verified by comparing the predicted stress–strain curves with experimental data and using it in finite element (FE) numerical simulation, respectively. A law of mixture of flow stress was also developed to analyse the strain and stress partitioning of 𝛿-ferrite and austenite of 2205 stainless steel, and AH36 carbon steel. The proposed constitutive model and the analysis of strain portioning could be used to predict the hot forming behaviour of 2205/AH36 BC under hot working conditions, providing a reference for the optimisation of the practical hot processing of bimetal composite. 2. Experimental procedure The bimetal composite employed in this study was a hot-rolled 2205/AH36 BC, which was produced with a high rolling reduction at 950 °C, followed by water cooling, and then annealed at 850 °C. The thickness values of 2205 layer and AH36 layer are 3 and 8 mm, respectively, as schematically shown in Fig. 1. The chemical compositions are listed in Table 1. As this study focuses on the deformation properties of composite during hot forming, the deformation temperatures should be selected to avoid the formation of sigma precipitation phase of 2205 duplex stainless steel, and the strain rates of deformation need to be relatively low [27]. Under this background, the hot tensile tests were conducted on a Gleeble 3500 thermal-mechanical test simulator over the temperature range of 950–1250 °C in steps of 100 °C, and at three strain rates of 0.01, 0.1 and 1 s−1 . Tensile specimens of bimetal composite with the thickness of 1.5 mm for each steel layer were cut by the Wire cut Electrical Discharge Machining (WEDM). The dimensions of tensile specimens are shown in Fig. 2a, and the Gleeble 3500 thermal-mechanical test simulator assembled with a tensile specimen is shown in Fig. 2b.

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International Journal of Mechanical Sciences 000 (2020) 105317

Table 1 Chemical compositions of the studied bimetal composite (wt%). Steel

C

Si

Mn

P

S

Cr

Mo

Ni

2205 AH36

0.024 0.18

0.62 0.50

1.4 0.90–1.60

0.023 0.035

0.001 0.035

21.07 0.20

3.01 0.08

5.36 0.40

Fig. 2. (a) Dimensions of tensile specimens, and (b) Gleeble thermal-mechanical test simulator assembled with a tensile specimen.

Prior to hot tensile tests, the specimens were reheated to 950, 1050, 1150 and 1250 °C, respectively, at a rate of 10 °C/s and held for 90 s to homogenise the temperature distribution. Tensile tests were then proceeded with the strain rates of 0.01, 0.1, and 1 s−1 , respectively, until the specimens were broken. Finally, the tested specimens were immediately quenched by water to avoid microstructure modifications during slow cooling.

3. Results and discussion 3.1. Hot flow behaviour The hot flow behaviour of bimetal composite depends on its peculiar microstructure. It has been found that the flow behaviour of 2205 duplex steel composed by a harder austenite phase and a softer ferrite phase at elevated temperatures is very similar and close to that of ferritic stainless steel which undergoes extensive DRV as a result of a high stacking fault energy (SFE) [26]. In contrast, the microstructure of AH36 carbon steel at high temperature is composed of full austenite which mainly softens by a vast DRX due to its low SFE [5]. Fig. 3 shows the flow stress–strain curves of the 2205/AH36 BC over the deformation temperature range from 950 to 1250 °C and the strain rates of 0.01, 0.1 and 1 s−1 . The obvious decrease of flow stress could be observed after it reaches the peak at a small strain, denoted as 𝜎 p , from flow curves with strain rates of 0.01 and 0.1 s−1 at all temperatures except 950 °C at the strain rate of 0.1 s−1 (Fig. 3a and b), and the peak stresses decrease with increasing the deformation temperature. According to these obtained curves, the hot deformation behaviour of 2205/AH36 BC is mainly controlled by the DRX softening mechanism at relative low strain rates (0.01 and 0.1 s−1 ). In addition, multiple peaks can be noticed from the stress curves at the strain rate of 0.01 s−1 , which is attributed to the occurrence of several successive cycles of DRX before the steady-state strain at relative low strain rates [28]. Compared with the curves at low strain rates, the flow stress reaches the peak value and keeps a stable value until necking begins at relative high strain rate of 1 s−1 for all the temperatures and at the strain rate of 0.1 s−1 at 950 °C (Fig. 3b and c), which means that the hot flow behaviour under these conditions is determined by the DRV softening mechanism [29]. Therefore, the softening mechanism (DRX or

DRV) of 2205/AH36 BC changes depending on the externally imposed working conditions, i.e., working temperatures and strain rates. On the basis of the feature of flow stress–strain curves, the flow stress controlled by the DRX softening mechanism can be further characterised by four distinctive stages [29,30]: (i) the stage I is related to the balance between dislocation storage and annihilation by WH and DRV, respectively; (ii) the stage II represents the progress of an additional softening mechanism (DRX) from the critical stress (𝜎 c ) to the steady state stress (𝜎 ss ); (iii) the stage III is characterised by a steady-state region until the necking occurs; and (iv) in the stage IV, the steady-state region begins to change due to the onset of necking, as shown in Fig. 3d. On the other hand, the flow stress controlled by the DRV softening mechanism is only characterised by the coexisting stage of WH and DRV (stage I). 3.2. Constitutive model A constitutive model was established from the experimental flow stress curves to describe and predict the flow behaviour of bimetal composite under hot-working conditions. For DRX softening mechanism, the model can be developed separately for preceding three stages (stages I–III) ignoring the stage IV which is determined by the damage evolution rules in FE numerical simulation instead of the constitutive model. In addition, the established model will be available for DRV softening mechanism in absence of DRX after removing the latter stages II and III. In the stage I, WH and DRV are the dominant mechanisms, and a KM model [31] was developed to predict the evolution of dislocation density (𝜌) with strain during deformation process, expressed as: d𝜌 = 𝑈 − Ω𝜌 d𝜀

(1)

where U represents the process of dislocation owing to WH, and Ω characterises the dislocation annihilation and rearrangement due to DRV. U and Ω are both considered independent of strain. Integrating Eq. (1) gives: 𝜌 = 𝜌0 𝑒−Ωε +

) 𝑈( 1 − 𝑒−Ωε Ω

(2)

where 𝜌0 is the initial dislocation density (when 𝜀 = 0) due to the deformation history or by nature [32], and 𝜀 is the strain. At high temperatures, the effective stress resulting from the short-range barriers,

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Fig. 3. True stress–strain curves of bimetal composite at the strain rate of (a) 0.01 s−1 , (b) 0.1 s−1 and (c) 1 s−1 over the temperature range from 950 to 1250 °C, and (d) the typical flow stress curve including WH and DRV stage, WH, DRV and DRX stage, DRX stage, as well as DRX and necking stage.

including the Peierls stress, and point effects (e.g., vacancies and selfinterstitials), can be negligible, so that the relationship between the flow stress (𝜎) and dislocation density is described as [11]: √ 𝜎 = 𝑀 𝛼0 𝜇𝑏 𝜌 (3) where M is the Schmidt orientation factor (3.06), 𝛼 0 is a proportional constant, 𝜇 is the shear modulus, and b is the magnitude of the Burgers vector (2.54 × 10−10 m). Then, substituting Eq. (2) into Eq. (3) yields the following expression: √ ( ) 𝜎 = 𝜎0 2 𝑒−Ωε + 𝜎𝑠 2 1 − 𝑒−Ωε (4) √ √ with 𝜎0 = 𝑀𝛼𝜇𝑏 𝜌0 and 𝜎𝑠 = 𝑀𝛼𝜇𝑏 𝑈 ∕Ω, where 𝜎 s is the saturation stress for DRV and its value is close to 𝜎 p (𝜎 s ≈ 𝜎 p ) in the presence of DRX. The assumption is given that the elastic deformation is negligible, i.e., the initial stress 𝜎 0 is supposed to be zero in this study, owing to the very beginning occurrence of plastic deformation [33], so that Eq. (4) becomes: √( ) 𝜎 = 𝜎𝑝 1 − 𝑒−Ωε (5) In order to determine 𝜎 p and Ω relevant to hot deformation conditions, a widely-used Arrhenius equation [11,34,35] was employed to describe the relationship between the flow stress, strain rate (𝜀̇ ) and

temperature, which is given by: ( ) 𝑄 𝑍 = 𝜀̇ exp = 𝐴[sinh (𝛼𝜎)]𝑛 𝑅𝑇

(6)

where Z is the Zener–Hollomon parameter [36], Q is the activation energy (kJ mol−1 ), T is the absolute deformation temperature (K), R is the gas constant (8.3145 J mol−1 K−1 ), and A and 𝛼 are material constants. The peak stress (𝜎 p ) was used as the input data for the regression process to determine the material constants (Q, 𝛼, A and n). According to Eq. (6), the activation energy (Q) can be determined by: } { { [ ( )] } 𝜕 ln sinh 𝛼𝜎𝑝 𝜕 ln 𝜀̇ 𝑄=𝑅 (7) [ ( )] 𝜕 (1∕𝑇 ) 𝜕 ln sinh 𝛼𝜎𝑝 𝑇 =constant

𝜀̇ =constant

The value of 𝛼 can be calculated as 0.01272 by the relationship of (𝜕 ln 𝜀̇ ∕𝜕 𝜎𝑝 )𝑇 ∕(𝜕 ln 𝜀̇ ∕𝜕 ln 𝜎𝑝 )𝑇 [37], and Q of bimetal composite can be determined as 628.9 kJ mol−1 . Based on Eq. (6), the correlation of ln Z versus ln [sinh (𝛼𝜎 p )] is obtained, as shown in Fig. 4a, to calculate the values of A (2.156 × 1022 ) and n (7.00), respectively, and the equation describing the relationship between 𝜎 p and Z is shown as: ( ) 𝜎𝑝 = 78.62 × sinh−1 0.000646 ⋅ Z0.143 (8a) and 𝑍 = 𝜀̇ exp

(

628900 RT

) (8b)

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International Journal of Mechanical Sciences 000 (2020) 105317

Fig. 4. (a) Relationship between peak stress (𝜎 p ) and Zener-Hollomon parameter (Z), and (b) relationship between peak strain (𝜀p ) and Zener–Hollomon parameter (Z).

Fig. 5. (a) Relationship between softening coefficient (Ω) and Zener–Hollomon parameter (Z), and (b) relationship between volume fraction of DRX (XD ) and strain under different DRX conditions.

Similarly, the strain at the position of peak strain (𝜀p ) can be expressed as a function of Z by the linear relationship of ln 𝜀p versus ln Z, as shown in Fig. 4b, and 𝜀p is denoted as: 𝜀𝑝 = 6.21 × 10−4 ⋅ Z0.052

(9)

Substituting the experimental results of stress and strain before critical strain for all tested situations into Eq. (5), the values of Ω can be obtained to establish the correlation between Ω and Z, as shown in Fig. 5a, given by: Ω = 29732.6 ⋅ Z−0.0896

(10)

The stage I can be employed to establish the constitutive model for the DRV softening mechanism. As mentioned in Section 3.1 that the imposed external working conditions, including temperatures and strain rates, affect the types of softening mechanism, so the value of ZenerHollomon parameter (Z) was adopted to determine the possible experienced softening mechanism for the 2205/AH36 BC. In Fig. 3b, the flow stress–strain curve feature is an exception at 950 °C compared to those at other temperatures with the strain rate of 0.1 s−1 . In this case, the Z value at 950 °C with the strain rate of 0.1 s−1 was set as the critical point to distinguish the DRV or DRX softening mechanism, and the Z value

can be calculated as 7.24158 × 1025 in this study. Therefore, it can be drawn that the DRV soften mechanism dominates the flow behaviour of 2205/AH36 BC when Z ≥ 7.24158 × 1025 . The DRX, otherwise, is the main softening mechanism when Z < 7.24158 × 1025 . The stages II and III are controlled by the DRX softening mechanism. The volume fraction of DRX (XD ), can be denoted as [34]: [ ( ) ] ) 𝜀 − 𝜀𝑐 𝑐 ( 𝜀 ≥ 𝜀𝑐 (11) 𝑋𝐷 = 1 − exp −𝑘 𝜀𝑝 where 𝜀c is the critical strain for the beginning of DRX, taken by 0.8𝜀p here [9], and k and c are material constants related to the mechanism of nucleation and DRX growth rate, respectively. Meanwhile, it is assumed that XD is proportional to the stress softening: 𝑋𝐷 =

𝜎𝑝 − 𝜎 𝜎𝑝 − 𝜎𝑠𝑠

𝜀 ≥ 𝜀𝑐

(12)

According to Eqs. (11) and (12), the flow stress under DRX softening conditions can be obtained by: { [ ( ) ]} ( ) ( ) 𝜀 − 𝜀𝑐 𝑐 𝜎 = 𝜎𝑝 − 𝜎𝑝 − 𝜎𝑠𝑠 1 − exp −𝑘 𝜀 ≥ 𝜀𝑐 (13) 𝜀𝑝

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International Journal of Mechanical Sciences 000 (2020) 105317

Fig. 6. (a) Original microstructure of bimetal composite, (b) schematic representation of the phase distribution at high temperatures, and (c) two imposed situations of 2205 stainless steel.

Taking logarithm of both sides of Eq. (11) twice, it becomes: ( )] 𝜀 − 𝜀𝑐 ln − ln 1 − 𝑋𝐷 = ln 𝑘 + 𝑐 ln 𝜀𝑝 [

(14)

The relationship between ln [ − ln (1 − XD )] and ln [(𝜀 − 𝜀c )/𝜀p ], as shown in Fig. 5b, is used to determine the values of k and c, which are 1.06 and 2.383, respectively. The stage III is the steady state when XD = 1, i.e., 𝜎 = 𝜎 ss , and 𝜎 ss can be described as a function of Z: ( ) 𝜎𝑠𝑠 = 78.62 × sinh−1 0.00184 ⋅ Z0.12

(15)

Therefore, the constitutive model of 2205/AH36 BC based on DRV or DRX softening mechanism, respectively, can be summarised as: √( )( ) ⎧ 1 − 𝑒−Ω𝜀 𝜀 < 𝜀𝑐 ⎪𝜎 = 𝜎 𝑝 ( ) ⎪ ⎪𝜎 = 𝜎 𝑝 𝜀 ≥ 𝜀 𝑐 ( ) ⎪ −1 0.143 if Z ≥ 7.24158 × 1025 (16a) ⎨𝜎𝑝 = 78.62 × sinh 0.000646 ⋅ Z ⎪ −0.0896 Ω = 29732 . 6 ⋅ Z ⎪ ( ) ⎪ 628900 ⎪𝑍 = 𝜀̇ exp ⎩ RT and √( ⎧ )( ) 1 − 𝑒−Ω𝜀 𝜀 < 𝜀𝑐 ⎪𝜎 = 𝜎 𝑝 ⎪ [ ]} { ( ) ⎪ ( ) ( ) 𝜀 − 𝜀𝑐 2.383 1 − exp −1.06 𝜀 ≥ 𝜀𝑐 ⎪𝜎 = 𝜎𝑝 − 𝜎𝑝 − 𝜎ss 𝜀 𝑝 ⎪ ⎪𝜎 = 78.62 × sinh−1 (0.000646 ⋅ Z0.143 ) ⎪ 𝑝 ⎪𝜎 = 78.62 × sinh−1 (0.00184 ⋅ Z0.12 ) ⎨ ss ⎪Ω = 29732.6 ⋅ Z−0.0896 ⎪ ⎪𝜀 = 6.21 × 10−4 ⋅ Z0.052 ⎪ 𝑝 ⎪𝜀 = 4.968 × 10−5 ⋅ Z0.052 ⎪ 𝑐 ( ) ⎪ 628900 ⎪𝑍 = 𝜀̇ exp RT ⎩

elicited under the same imposed hot working conditions. It is clear that the 2205 stainless steel and AH36 carbon steel deformed in parallel with the same strain rate in the hot tensile process, which can be described by the mixture law: 𝜎𝑡𝑜𝑡𝑎𝑙 = 𝑓2205 𝜎2205 + 𝑓𝐴𝐻36 𝜎𝐴𝐻36

(17a)

and 𝜀̇ 𝑡𝑜𝑡𝑎𝑙 = 𝜀̇ 2205 = 𝜀̇ 𝐴𝐻36

(17b)

where 𝜎 total , 𝜎 2205 , 𝜎 AH36 , 𝜀̇ 𝑡𝑜𝑡𝑎𝑙 , 𝜀̇ 2205 and 𝜀̇ 𝐴𝐻36 are the stress of bimetal composite, stress of 2205 stainless steel, stress of AH36 carbon steel, strain rate of bimetal composite, strain rate of 2205 stainless steel and strain rate of AH36 carbon steel, respectively. The f stands for the volume fraction of each material, and f2205 + fAH36 = 1. The stress partitioning exists between 𝜎 2205 and 𝜎 AH36 which are not equivalent when the strain rate and strain accumulation are the same under all the conditions. Moreover, as there are two phases with different strengths in the 2205 stainless steel layer, the flow stress and strain vary from phase to phase. A law of mixture can be adopted to describe the flow behaviours of dual phase steels by the following equations [26]: 𝜀̇ 2205 = 𝑓𝛾 𝜀̇ 𝛾 + 𝑓𝛿 𝜀̇ 𝛿

(18a)

and 𝜎2205 = 𝑓𝛾 𝜎𝛾 + 𝑓𝛿 𝜎𝛿

(16b)

if Z < 7.24158 × 1025 3.3. Strain and stress partitioning In the previous section, the total hot flow behaviours and constitutive model of 2205/AH36 BC have been discussed. However, in case of the coexistence of two material layers (Fig. 6a), it is interesting to consider the contribution of each material on the total performance of bimetal composite. The microstructure of AH36 carbon steel layer is with prior austenite grains (PAGs) in the experimental temperature range of 950– 1250 °C that are higher than the austenitising temperature of carbon steel, and the 2205 stainless steel consists of a certain amount of 𝛿-ferrite and austenite at different deformation temperatures. The schematic representation of the phase distribution is shown in Fig. 6b. As the hot straining response between two different materials of bimetal composite, as well as austenite and 𝛿-ferrite in 2205 stainless steel, is extremely different, the stress and strain partitionings are

(18b)

where indices 𝛿 and 𝛾 refer to 𝛿-ferrite and austenite, respectively, and f𝛾 + f𝛿 = 1. During the hot deformation process of 2205 stainless steel, the softer 𝛿-ferrite tends to deform preferentially compared to the harder austenite phase which even impedes the deformation at the early beginning of tests, but the effective load will be progressively transferred from 𝛿-ferrite to austenite at the later stage, in which the strain rate in 𝛿-ferrite gradually declines toward the testing strain rate, and the austenite starts to deform and accommodate more strain deformation [24]. Therefore, two extreme cases of I and II, as shown in Fig. 6c, can be considered during the process of dynamic strain rates: 𝛿-ferrite and austenite are in series (iso-stress condition), and they deform in parallel (iso-strain condition), respectively [38]. Thus, for iso-stress condition that the 𝛿-ferrite gives the highest contribution of deformation to 2205 stainless steel, it is reasonably supposed that: 𝜀̇ 𝛾 ≈ 0 and 𝜀̇ 𝛿 = Ω𝛿 𝜀̇ 2205

(19)

where Ω𝛿 denotes the contribution coefficient of 𝛿-ferrite, and (𝜀̇ 𝛿 )max = 1∕(1 − 𝑓𝛾 )𝜀̇ 2205 in the absence of any deformation of austenite, but the austenite component doses not undergo absolute zero-strain in the real situations so that assuming (Ω𝛿 )max = 0.98/(1 − f𝛾 ). In contrast, the 𝛿ferrite gives the lowest contribution of 𝜀2205 under the iso-strain condition, namely: ( ) (20) 𝜀̇ 𝛿 min = 𝜀̇ 𝛾 = 𝜀̇ 2205 As the phase morphology is more complicated than either of extreme cases in reality, the contribution coefficients are adopted in the real

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International Journal of Mechanical Sciences 000 (2020) 105317

mixture conditions, and the following equations for the strain rate of each phase are proposed: ( ) 𝜀̇ 𝛿 = Ω𝛿 𝜀̇ 2205 1 ≤ Ω𝛿 ≤ 0.98∕(1 − 𝑓𝛾 ) (21a) and

( ) 𝜀̇ 𝛾 = Ω𝛾 𝜀̇ 2205 0.02∕𝑓𝛾 ≤ Ω𝛾 ≤ 1

(21b)

where, Ω𝛾 denotes the contribution coefficient of austenite (Ω𝛾 = [1 − (1 − f𝛾 )Ω𝛿 ]/f𝛾 ). The variation of Ω𝛿 can be described as a sigmoidal function [24,26]: ( ) 0.98 1∕𝑓𝛿 − 1 Ω𝛿 = 1 + (22) ( )𝑞 1 + 𝜀2205 ∕0.5𝜀𝑡 where 𝜀2205 and 𝜀t are the strain of 2205 stainless steel and total strain required to produce the transition at which load is transferred from 𝛿ferrite to austenite, respectively, and q is a constant. However, it is difficult to determine the effective amount of straining in 𝛿-ferrite, i.e., the value of Ω𝛿 , during tests, since the value depends not only on the volume fraction of phases, but also on the magnitude of 𝜀t . For simplicity, it can be calculated by the regression method at low strains for the hot tensile tests based on the experimental data, which will be described later. Thus, based on Eqs. (17b), (18b) and (21), the mixture law of total stress for the bimetal composite (Eq. (17a)) can be rewritten as: ( ) 𝜎total = 𝑓2205 𝑓𝛾 𝜎𝛾 ⟨Ωγ ε̇ total ⟩ + 𝑓𝛿 𝜎𝛿 ⟨Ω𝛿 𝜀̇ total ⟩ + 𝑓AH36 𝜎AH36 ⟨𝜀̇ total ⟩ (23) where 𝜎𝛾 ⟨Ωγ ε̇ total ⟩, 𝜎𝛿 ⟨Ω𝛿 𝜀̇ total ⟩ and 𝜎AH36 ⟨𝜀̇ total ⟩ denote the function of the total strain rate (𝜀̇ 𝑡𝑜𝑡𝑎𝑙 ), i.e., the testing strain rate, respectively. In addition, it was assumed that the flow stress remains constant after the peak in absence of DRX here. Based on Arrhenius equation, the experimental relationships between strain rate and stress in the peak stress state for single phase ferritic and single phase austenitic stainless steels are, respectively, given by [39]: ( ) [ ( )]3.64 310000 𝜀̇ 𝛿 = 6.32 × 1012 sinh 0.0103𝜎𝑝𝛿 exp − (24a) RT and

( ) [ ( )]4.57 454000 𝜀̇ 𝛾 = 1.44 × 1015 sinh 0.007𝜎𝑝𝛾 exp − RT

(24b)

The peak stress of 𝜎 AH36 from individual hot tensile tests of AH36 carbon steel can be calculated by: ( ) 𝜎𝑝𝐴𝐻36 = 55.6 × sinh−1 0.015 ⋅ Z𝐴𝐻36 0.22 (25a) and 𝑍AH36 = 𝜀̇ exp

(

273000 RT

) (25b)

In addition, Thermocalc software was employed to obtain the relationship between the austenite volume fraction and deformation temperature, namely: 𝑓𝛾 = −0.0012𝑇 + 2.14

(26)

where T is the absolute deformation temperature (K). Then, Ω𝛿 can be calculated as: ⎧1 ⎪ Ω𝛿 = ⎨0.1376 ⋅ Z𝛿 0.1041 ( ) ⎪ ⎩0.98∕ 1 − 𝑓𝛾 and 𝑍𝛿 = 𝜀̇ exp

(

310000 RT

Fig. 7. Predicted strain rate ratios of 𝛿-ferrite to austenite, 𝛿-ferrite to composite, and austenite to composite.

1 s−1 at different testing temperatures, respectively, as shown in Fig. 7. It is found that the predicted strain partitioning of two phases in 2205 stainless steel decreases with the increase of temperature, i.e., 𝜀̇ 𝛿 ∕𝜀̇ 𝛾 decreases with temperature. It can be explained that the less volume fraction of existing austenite needs to share more deformation strains with the increase of temperature. Moreover, the value of 𝜀̇ 𝛿 ∕𝜀̇ 𝛾 is large but that of 𝜀̇ 𝛾 ∕𝜀̇ 𝑡𝑜𝑡𝑎𝑙 is quite small, meaning that the contribution of austenite on the total strain rate is little, especially at low total strain rates where the net strain rate values of austenite are quite small. In addition to the effect of working temperatures, the strain rates have less influence on the value of 𝜀̇ 𝛿 ∕𝜀̇ 𝑡𝑜𝑡𝑎𝑙 but more effect on the value of 𝜀̇ 𝛿 ∕𝜀̇ 𝑡𝑜𝑡𝑎𝑙 at relative highly temperatures, which indicates that the austenite of 2205 stainless steel layer is more sensitive to strain rates than the 𝛿-ferrite. Due to the existence of stress partitioning between AH36 carbon steel and 2205 stainless steel, the predicted peak stress ratios of AH36 carbon steel to 2205 stainless steel (𝜎 pAH36 /𝜎 p2205 ), AH36 carbon steel and to bimetal composite (𝜎 pAH36 /𝜎 total ), and 2205 stainless steel to bimetal composite (𝜎 p2205 /𝜎 total ) are obtained under different hot tensile conditions, respectively, as shown in Fig. 8. It is noted that the values of 𝜎 pAH36 /𝜎 p2205 and 𝜎 pAH36 /𝜎 total increase (Fig. 8a and b) but that of 𝜎 p2205 /𝜎 total decreases (Fig. 8c) with the increase of temperatures, indicating that the contribution of AH36 carbon steel layer to total stress is relatively more than that of 2205 stainless steel layer at high temperatures. A reason is that the soft 𝛿-ferrite phase in 2205 stainless steel is transformed from the hard austenite at high temperatures, and the stress of 𝛿-ferrite is less when the same strain is imposed.

if Ω𝛿 < 1

( ) if 1 ≤ Ω𝛿 ≤ 0.98∕ 1 − 𝑓𝛾 ( ) if Ω𝛿 > 0.98∕ 1 − 𝑓𝛾

(27a)

) (27b)

The calculation process of Ω𝛿 is given in the Appendix A when 1 ≤ Ω𝛿 ≤ 0.98/(1 − f𝛾 ). Consequently, the strain rate ratios of 𝛿-ferrite to austenite (𝜀̇ 𝛿 ∕𝜀̇ 𝛾 ), 𝛿-ferrite to composite (𝜀̇ 𝛿 ∕𝜀̇ 𝑡𝑜𝑡𝑎𝑙 ), and austenite to composite (𝜀̇ 𝛾 ∕𝜀̇ 𝑡𝑜𝑡𝑎𝑙 ) can be obtained with the total strain rates of 0.01, 0.1 and

3.4. Comparison between constitutive model and mixture law The hot tensile tests of 2205/AH36 BC at 1000 °C with the strain rates of 0.01, 0.1 and 1 s−1 , which do not belong to the training data groups for the constitutive model, were carried out to compare the prediction results of the proposed constitutive model with that of the mixture law (Eq. (23)). Fig. 9 shows the experimental stress–strain curves up to the strain of 0.08 before the onset of necking at 1000 °C with the strain rates of 0.01, 0.1 and 1 s−1 , predicted stress–strain curves from the constitutive model, and predicted peak stresses from the mixture law. It can be observed that the predicted stress–strain curves by the consti-

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International Journal of Mechanical Sciences 000 (2020) 105317

Fig. 8. Predicted peak stress ratios of (a) AH36 carbon steel to 2205 stainless steel, (b) AH36 carbon steel to bimetal composite, and (c) 2205 stainless steel to bimetal composite.

Fig. 9. Comparison for the experimental strain–stress curves, predicted curves from the proposed constitutive model, and predicted peak stresses at 1000 °C (up to the strain of 0.08).

tutive model agree well with the experimental data, thus verifying the prediction accuracy of the proposed constitutive model in this study. Meanwhile, the predicted peak stresses of the mixture law at the strain rates of 0.1 and 1 s−1 are close to the experimental saturation stress, respectively. As the assumption of mixture law cannot take into account the effect of DRX, there is a certain gap between the predicted peak stress of the mixture law and final experimental steady state stress at the strain rate of 0.01 s−1 . The ultimate purpose of the constitutive model is to be used for numerical simulation to predict the flow behaviour of bimetal composite. Based on the commercial software ABAQUS, FE simulation model was established by using the proposed constitutive model and different material arrangement schemes considering the strain partitioning of bimetal composite, respectively, as shown in Fig. 10. The first scheme is that the constitutive model was assigned to the entire gauge length zone (Fig. 10a). For others, the material property of AH36 (Eq. (25)) was individually assigned to the gauge length zone of the AH36 carbon steel layer, and the material properties of 𝛿-ferrite as well as austenite were assigned to corresponding zones according to three types of arrangement schemes along the longitudinal section, i.e., 𝛿-ferrite and austenite in parallel (Fig. 10b), 𝛿-ferrite and austenite in series (Fig. 10c), as well as arbitrary 𝛿-ferrite and austenite (Fig. 10d). The volume fraction of each phase can be calculated by Eq. (26), and it was assumed that

Fig. 10. FE simulation arrangement schemes: (a) proposed constitutive model, (b) 𝛿-ferrite and austenite in parallel, (c) 𝛿-ferrite and austenite in series, and (d) arbitrary 𝛿-ferrite and austenite.

the temperature of each phase remains constant at 1000 °C during the simulation process in this study. The peak loads from hot tensile test at 1000 °C with the strain rate of 1 s−1 , FE simulation by using the constitutive model, and three types of arrangement schemes are shown in Fig. 11a. It is clear that the peak loads obtained from the constitutive model and the scheme of 𝛿-ferrite and austenite in parallel are close to the experimental result, and the difference between the FE and experimental results is only 7.6% and 10.3%, respectively. However, the FE results calculated by the other two arrangement schemes, i.e., parallel and arbitrary 𝛿-ferrite and austenite, are quite lower than the experimental peak load (approximately 24% reduction). This is due to the strain partitioning between 𝛿-ferrite and austenite, resulting in the occurrence of partial necking in softer 𝛿-ferrite zones in advance, as shown in Fig. 11b and c. As the FE simulation model is almost impossible to be completely established according to the real microstructure in which all the phases are intertwined, it is unable to avoid the partial necking. As a result of the limit of FE simulation model, the FE results are relative lower than the practical data. Consequently, the proposed constitutive model of entire material tends to obtain closer predicted results in order to calculate the hot forming load or rolling

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International Journal of Mechanical Sciences 000 (2020) 105317

Fig. 11. (a) Peak loads obtained from experiment, FE simulation by using the constitutive model, as well as parallel, serial and arbitrary 𝛿-ferrite and austenite at 1000 °C with the strain rate of 1 s−1 , respectively, (b) stress distribution of 𝛿-ferrite and austenite in series, and (c) stress distribution of arbitrary 𝛿-ferrite and austenite.

force of bimetal composite based on FE simulation method in practical applications.

Declaration of Competing Interest The authors declare that there is no conflict of interest.

4. Conclusions

Acknowledgments

Based on the hot tensile tests over the temperature range of 950– 1250 °C and strain rate range of 0.01–1 s−1 , the flow behaviour and strain partitioning of 2205/AH36 BC were investigated in this study. The main conclusions of this research are drawn as follows:

The first author is greatly thankful for the scholarship support (IPTA) from University of Wollongong and scholarship support from China Scholarship Council (CSC) (grant no. 201608430108).

1 The unique hot flow behaviour of 2205/AH36 BC differs from that of a single material such as either the AH36 carbon steel or the 2205 stainless steel. The softening mechanisms (DRV or DRX) of 2205/AH36 BC changes depending on the externally imposed temperatures and strain rates. When the Zener–Hollomon parameter related to working temperatures and strain rates is greater than a certain value (7.24158 × 1025 in this study), the 2205/AH36 BC shows DRV softening mechanism. Conversely, DRX occurs during the hot deformation process of 2205/AH36 BC. 2 According to the experimental stress–strain curves, a physicallybased constitutive model considering DRV and DRX softening mechanisms of 2205/AH36 BC was established. The constitutive model was verified by the experimental data at 1000 °C with strain rates of 0.01, 0.1, 1 s−1 . The predicted stress–strain curves by the constitutive model are close to practical results from hot tensile tests. 3 A mixture law of bimetal composite considering the strain and stress partitioning was developed to calculate the total stress of 2205/AH36 BC. It is found that the working temperatures have the effect on the strain partitioning between the austenite and 𝛿-ferrite of 2205 stainless steel, and the austenite phase is more sensitive to strain rates than the 𝛿-ferrite at relative high temperatures. The contribution of AH36 carbon steel layer on total stress is relatively more than that of 2205 stainless steel layer at high temperatures, due to the occurrence of stress partitioning. 4 Based on FE numerical simulation, the calculated peak loads obtained from the constitutive model are close to the experimental results, but the predicted loads from assigning material properties to each zone are less due to the strain partitioning. For the practical applications, the proposed constitutive model can be employed to meet the calculation requirements.

Appendix A. The calculation process of 𝛀𝜹 , when 1 ≤ 𝛀𝜹 ≤ 0.98/(1 − f𝜸 ) According to the stress–strain curves from hot tensile tests, as shown in Fig. 3, the peak stresses of 2205/AH36 BC are provided in Table A1. The peak stresses of AH36 carbon steel were obtained by individual tensile tests under the same hot working conditions, as shown in Table A2. It was assumed that the flow stress remains constant after the peak in absence of DRX, and the strain at which AH36 carbon steel as well as the austenite and 𝛿-ferrite of 2205 stainless steel reach the peak stress is equal. Therefore, based on Eq. (23), the peak stress of 2205/AH36 BC can be rewritten as: ) ( 𝜎ptotal = 𝑓2205 𝑓𝛿 𝜎𝑝𝛿 ⟨Ωδ ε̇ total ⟩ + 𝑓𝛾 𝜎𝑝𝛾 ⟨Ω𝛾 𝜀̇ total ⟩ + 𝑓AH36 𝜎pAH36 ⟨𝜀̇ total ⟩ (A1)

Table A1 The peak stresses of 2205/AH36 BC (𝜎 ptotal ) under hot tensile conditions (MPa). Strain rate Stress Temperature

0.01 s−1

0.1 s−1

1 s−1

950 °C 1050 °C 1150 °C 1250 °C

102 66.9 45 32

151 98 59 44

185 101 82 56

Table A2 The peak stresses of AH36 carbon steel (𝜎 pAH36 ) under hot tensile conditions (MPa). Strain rate

Funding This research received financial support from Baosteel-Australia Joint Research and Development Centre(BAJC) under project of BA16009.

Stress Temperature

0.01 s−1

0.1 s−1

1 s−1

950 °C 1050 °C 1150 °C 1250 °C

80 59 44 33

106 83 65 50

134 110 90 73

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International Journal of Mechanical Sciences 000 (2020) 105317

The peak stresses of 𝛿-ferrite and austenite of 2205 stainless steel can be calculated by Eq. (24), namely: [ ] ( ( )) 310000 0.275 𝜎𝑝𝛿 = 97.08 × sinh−1 0.0003 Ω𝛿 𝜀̇ total exp (A2) RT and [ ] ( ( )) 454000 0.219 𝜎𝑝𝛾 = 142.85 × sinh−1 0.00048 Ω𝛾 𝜀̇ total exp RT

(A3)

The relationships of volume fraction and contribution coefficients are given by: 𝑓2205 = 𝑓𝐴𝐻36 = 0.5

(A4)

𝑓𝛾 = −0.0012𝑇 + 2.14

(A5)

𝑓𝛿 = 1 − 𝑓𝛾

(A6)

and

[ ] Ω𝛾 = 1 − (1 − 𝑓𝛾 )Ω𝛿 ∕𝑓𝛾

(A7)

Substituting Eqs. (A2)–(A7) into (A1) can be obtained: ( 𝜎𝑝𝑡𝑜𝑡𝑎𝑙 = 0.5 (0.0012𝑇 + 1.14) × 97.08 ( ) ( ( )) 310000 0.275 × sinh−1 0.0003 Ω𝛿 𝜀̇ 𝑡𝑜𝑡𝑎𝑙 exp 𝑅𝑇 + (−0.0012𝑇 + 2.14) × 142.85 × sinh−1 ( )) ( ( ))0.219 1 − (1.14 + 0.0012𝑇 )Ω𝛿 454000 0.00048 𝜀̇ 𝑡𝑜𝑡𝑎𝑙 exp −0.0012𝑇 + 2.14 𝑅𝑇 + 0.5𝜎𝑝𝐴𝐻36 𝜀̇ 𝑡𝑜𝑡𝑎𝑙

(A8)

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