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A physically-based constitutive modelling of a high strength aluminum alloy at hot working conditions
Accepted Manuscript A physically-based constitutive modelling of a high strength aluminum alloy at hot working conditions Hongming Zhang, Gang Chen, Qiang Chen, Fei Han, Zude Zhao PII:
S0925-8388(18)30469-9
DOI:
10.1016/j.jallcom.2018.02.039
Reference:
JALCOM 44915
To appear in:
Journal of Alloys and Compounds
Received Date: 30 October 2017 Revised Date:
4 February 2018
Accepted Date: 5 February 2018
Please cite this article as: H. Zhang, G. Chen, Q. Chen, F. Han, Z. Zhao, A physically-based constitutive modelling of a high strength aluminum alloy at hot working conditions, Journal of Alloys and Compounds (2018), doi: 10.1016/j.jallcom.2018.02.039. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A physically-based constitutive modelling of a high strength aluminum alloy at
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hot working conditions
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Hongming Zhang a, Gang Chen b,*, Qiang Chen c,*, Fei Han b, Zude Zhao c
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a
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China
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b
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Weihai 264209, China
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c
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China
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Department of Civil Engineering, Harbin Institute of Technology, Weihai 264209,
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School of Materials Science and Engineering, Harbin Institute of Technology,
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Southwest Technology and Engineering Research Institute, Chongqing 400039,
* Corresponding author. E-mail address: [email protected] (Gang Chen); [email protected] (Qiang Chen)
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Abstract: The hot deformation behavior of a high strength aluminum alloy
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(Al-Zn-Mg-Cu) was studied by isothermal hot compression tests performed over a
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range of temperatures (350~490 °C) and strain rates (0.001~1 s-1). A constitutive
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equation was established using experimental results to predict the flow stress of the
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alloy under elevated temperature. In the work hardening-dynamic recovery regime, a
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physically-based constitutive equation for the flow stress was obtained from the
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stress-dislocation relation. In the subsequent dynamic recrystallization region, the
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flow stress after the peak was predicted by employing the kinematics of the dynamic
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recrystallization in the constitutive model. The stress-strain curves of the alloy
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predicted by the established models were in good agreement with experimental results.
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The results indicate that the proposed physically-based constitutive equation can
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accurately predict the flow behavior of the Al-Zn-Mg-Cu alloy.
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Keywords: Al-Zn-Mg-Cu alloy; Hot deformation behavior; Dynamic recovery and
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dynamic recrystallization; Physically-based constitutive model
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1. Introduction Al-Zn-Mg-Cu alloys belong to the high strength aluminum alloys, which have been
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widely used to manufacture airplane structures and auto parts due to high
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strength-to-weight ratio and good fatigue resistance properties. Due to lower plasticity
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of Al-Zn-Mg-Cu alloys at room temperature, hot plastic forming is usually chosen to
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process the materials. During the hot deformation process, the microstructures of the
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material change significantly because of the influence of temperatures, strains and
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strain rates [1, 2]. Furthermore, the microstructural evolution strongly affects the flow
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stress behavior [3, 4]. To design the thermomechanical processing parameters for
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Al-Zn-Mg-Cu alloys, it is necessary to study their hot deformation behavior.
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The constitutive equations, which establish correlations with the flow stress, strain rate and temperature, are often used to estimate the flow behavior of metal materials
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at hot working conditions. The stress-strain curves obtained by the uniaxial hot
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compression tests are often employed to provide the necessary data for the
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constitutive equations. In the previous investigations, various models have been
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proposed to predict the flow behavior of metal materials. Sellars and Tegart proposed
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the phenomenological approach in which the flow stress was described by the
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hyperbolic-sine Arrhenius equation [5], and much work has been done to study the
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flow behavior of various metal materials by the phenomenological model based on the
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hyperbolic-sine law [6-9]. The phenomenological model is rather straightforward,
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ACCEPTED MANUSCRIPT where only a few material constants are used to describe the constitutive equation.
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However this method does not fully consider the microstructural effects, and lacks
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physical meaning. In the plastic deformation, the dislocation density variation is one
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of the most important microstructural parameters. Based on the dislocation density
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variation, the physical constitutive model can be built which correlates with the
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physical mechanism of the work hardening (WH), dynamic recovery (DRV) and
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dynamic recrystallization (DRX). A number of physically-based constitutive models
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have been established to describe the hot deformation behavior of metal materials,
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such as 42CrMo steel [10], 55SiMnMo bainite steel [11], nickel-based super-alloy
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[12], Cu-Mg alloy [13], Ti6Al4V alloy [14], LDX 2101 duplex stainless steel [15],
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and 2050 Al-Li alloy [16]. These studies make accurate predictions for the
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experimental results, indicating that the physically-based constitutive models can
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accurately describe the flow stress of the metal materials at hot deformation
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conditions. Generally, an Al-Zn-Mg-Cu alloy has high stacking fault energy, and the
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insoluble dispersed phases exist in the alloy [17]. These factors seriously affect the
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dynamic recrystallization behavior during the hot deformation of Al-Zn-Mg-Cu alloys.
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Therefore, it is necessary to develop an accurate physically-based constitutive model
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to properly account for the intricate effect of hot working conditions on the flow
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behavior of Al-Mg-Zn-Cu alloys.
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The objective of the present work is to investigate the hot deformation
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characteristics of an Al-Zn-Mg-Cu alloy by utilizing the uniaxial hot compression
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tests and to establish a physically-based constitutive model to describe the flow stress
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of the alloy during the work hardening-dynamic recovery and dynamical
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recrystallization periods. Then the predictive performance of the developed
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constitutive model is discussed by comparing with the experimental data.
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2.
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Experimental procedure A high-strength aluminum alloy with a composition of Al-5.65Zn-2.18Mg-1.71Cu
(wt. %) was adopted for this work. The studied Al-Zn-Mg-Cu alloy was first extruded
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at 400 °C with an extrusion ratio of 17 and then machined to compression specimens
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with diameters of 8 mm and heights of 12 mm.
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According to the previous study by the authors regarding the plastic deformation characteristics of Al-Zn-Mg-Cu alloys in a warm deformation field [18], the hot
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compression tests were carried out in a temperature range of 350~490 °C and a strain
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rate range of 0.001~1 s-1 on a Gleeble 1500 thermomechanical simulator, which can
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automatically record true stress-true strain data. To minimize the friction between the
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specimens and the die during hot deformation, the flat ends of the specimen were
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recessed to entrap the high-temperature graphite lubricant. The thermocouple wires
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were welded onto the surface of the hot compressed specimen to measure the
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temperature of the specimens. The specimens were heated to a preset temperature at a
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rate of 10 °C/s, soaked for 3 min to ensure a uniform temperature over the entire
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specimen. After each compression test, the deformed specimens were quenched in
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water quickly to obtain the hot deformation microstructures. The microstructures of
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hot compressed specimen were examined by the Olympus PMG3 optical microscope
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and a JEOL-2100 transmission electron microscope.
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3.
Results and discussion
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3.1 Microstructural evolution
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Fig. 1 shows the initial microstructure of the extruded Al-Mg-Zn-Cu alloy. The results indicate the grains were fibrous and DRX did not occur in the microstructures.
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Moreover, the second phases were lath-like and distributed along the extrusion
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direction.
Fig. 2 shows the microstructures of the studied Al-Zn-Mg-Cu alloy compressed at a
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strain rate of 0.01 s-1 under different temperatures. As shown in Fig. 2 (a) and (b), the
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equiaxial recrystallized microstructures were not found after compression at 350 °C
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and 400 °C. When the deformation temperature was increased to 450 °C, some fine
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recrystallized grains were detected in the microstructures (Fig. 2(c)), indicating DRX
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had occurred during deformation. As the deformation temperature increased, a large
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amount of recrystallized grains appeared at 470 °C (Fig. 2(d)). It is assumed that
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higher temperatures can activate DRX and alter the softening mechanism [19].
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Fig. 3 shows the microstructures of the studied Al-Zn-Mg-Cu alloy compressed at
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470 °C under different strain rates. Fig. 3(a) exhibits the microstructures at a higher
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strain rate of 1 s-1, and the results demonstrate that only a few microstructures had
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been recrystallized. When the alloy was compressed at lower strain rates (0.1 s-1 and
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0.01 s-1), the recrystallized microstructures increased gradually. It is assumed that
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DRX is the predominant softening mechanism, whereas the softening caused by DRV
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is weak (Fig. 3(b) and (c)). When the alloy was deformed at a strain rate of 0.001 s-1,
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the microstructures consisted of subgrains and fine recrystallized grains, indicating
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ACCEPTED MANUSCRIPT that full DRX was attained during deformation (Fig. 3(d)). Therefore, the strain rate
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plays an important role in DRV and DRX during hot deformation. At lower strain
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rates, the dislocations have sufficient time to climb or slip into subgrain boundaries,
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and when the low-angle grain boundaries have transformed to large-angle grain
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boundaries, most of the subgrains can evolve into recrystallized grains [20].
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Fig. 4 shows the TEM micrographs of the studied Al-Zn-Mg-Cu alloy compressed
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at 350 °C /0.001 s-1, 470 °C /0.1 s-1 and 470 °C /0.001 s-1. As shown in Fig. 4(a), when
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the alloy was compressed at 350 °C/0.001 s-1, the dislocation underwent a multilateral
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movement based on the dislocation climb and slip, which occurred during the DRV
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stage and effectively reduced the dislocation density [21]. When the alloy was
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compressed at 470 °C/0.1 s-1, dislocation dissociation was detected, indicating that the
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nucleation caused by subgrain merging occurred and DRX participated in the hot
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deformation (Fig. 4(b)) [2]. When the alloy was compressed at 470 °C/0.001 s-1, the
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grains with typical DRX characteristics can be determined according to the clean and
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straight high-angle grain boundaries (Fig. 4(c)) [22,23]. The specifics of the
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microstructures in Fig. 4 suggest a conclusion that DRX effect works at higher
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temperatures and with lower strain rates.
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Haghdadi et al. recently reported a distinct softening mechanism analogous to
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discontinuous dynamic recrystallization within ferrite at a high strain rate [24].
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However, the stacking fault energy of an aluminum alloy is quite high resulting in
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recrystallization difficulties [17], and recrystallization time is insufficient at high
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strain rates. Therefore, the opposite tendency appears compared with that described in
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the report by Haghdadi et al.
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3.2 Hot deformation behavior
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The true stress-strain curves of the studied Al-Zn-Mg-Cu alloy during hot compression are shown in Fig. 5. It can be found that the peak stress increases as the
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deformation temperature decreases and the strain rate increases. This finding is
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because the higher strain rate causes more tangled dislocation structures, which hinder
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the dislocation movement and increase the resistance of the material deformation [9
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25]. Moreover, lower temperatures slow down thermally activated processes, resulting
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in insufficient motion of the dislocations and vacancies [19]. Furthermore, higher
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strain rates can reduce the coarsening time of dynamically recrystallized grains, and
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the lower temperatures inhibit the mobility of the recrystallized grain boundaries,
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which also lead to increasing the flow stress [26].
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As indicated in Fig. 5, the stress-strain curves can be divided into two distinct types, and the schematic curves marked with ‘a’ and ‘b’ in Fig. 6 indicate the different
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characteristics between two types of stress-strain curves. The characteristic of the
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stress curve marked with ‘a’ is attributed to DRV as the main softening mechanism. At
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the beginning of the deformation, the flow stress rapidly increases with an increase in
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strain due to the work hardening. When a balance between work hardening and
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dynamic recovery is reached, a saturation value (σsat) appears and remains constant.
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For the curve marked with ‘b’, it is considered that DRX is the dominant softening
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mechanism. The stress-strain curve marked with ‘b’ consists of three stages: stage I
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(work hardening stage), stage II (softening stage) and stage III (steady stage). In stage
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ACCEPTED MANUSCRIPT I, the flow stress rapidly increases to a critical stress (σc) at the initial deformation
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stage due to the work hardening. Once the deformation value exceeds the critical
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strain (εc), DRX occurs and the flow stress moves into stage II. It can be found that
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the flow stress continues to slowly increase to the peak stress (σp) because of the
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contest between the work hardening and dynamic softening mechanisms (including
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DRV and DRX). When the deformation exceeds the peak strain (εp), the dynamic
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softening caused by DRX becomes strong enough to overcome the work hardening,
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then the flow stress begins to decrease until it reaches a steady stress (σss). During
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stage III, a balance between work hardening and dynamic softening is reached, and
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the flow stress remains unchanged regardless of the increased strain. In Fig. 6, σrec
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represents the saturation stress during the steady-state deformation due to DRV, and
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σdrx represents the steady-state stress due to DRX. Assuming that DRX does not
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happen in the curve marked with ‘a’, the difference (∆σ) between σdrx and σrec is the
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net softening attributable to DRX.
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3.3 Work hardening behavior
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It is important to analyze the work hardening behavior of the alloy, which can
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clearly reflect the evolution in DRV. The relation between the strain-hardening rate
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(θ=dσ/dε) and the flow stress has been widely used to represent the work hardening
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behavior of alloys. Fig. 7 shows that the θ-σ curve of the studied Al-Zn-Mg-Cu alloy
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is compressed at different temperatures and strain rates. As shown in Fig. 7 (a), the
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work hardening rate decreases as the strain rate decreases and the temperature
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increases. During plastic deformation, the work hardening rate depends on the
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competition between the dislocation storage and annihilation, and the effect of DRV is
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enhanced as the strain rate decreases and the temperature increases.
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The fitting θ-σ curve at 470 °C/0.001 s-1 is shown in Fig. 7 (b). Initially, the value of θ decreases with an increase in stress, due to the softening effect induced by DRV.
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Then, an inflection appears in the θ-σ curve, which indicates that DRX occurs under
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the deformation conditions. Moreover, the inflection point in the θ-σ curve is
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generally used to determine the value of σc [13]. When the strain-hardening rate θ
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becomes 0, σp is achieved. The stress continuously decreases as the strain increases
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until the stress reaches σss, when the value θ becomes zero again. The horizontal
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intercept of the tangent line of the θ-σ curve through the inflection point can be
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regarded as σsat.
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3.4 Dynamic recovery modelling
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The variation in flow stress during hot deformation, which is under the control of
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work hardening and dynamic recovery is primarily attributed to the evolution in the
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dislocation density with strain. It is generally considered that the work hardening
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increases the number of dislocations and the dynamic recovery gives rise to
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dislocation annihilation and rearrangement. The evolution in the dislocation density
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with strain can be expressed as the following two components [27]:
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where ρ is the dislocation density, ε is the strain, dρ/dε is the increasing rate of the
d ρ / d ε = U − Ωρ
(1)
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dislocation density with strain; and U and Ω are the parameters with respect to the
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strain that represent the work hardening and softening terms, respectively. The
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dislocation density can be defined by integrating Eq. (1) as follows:
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ρ = e − Ωε U / Ω e Ωε + ρ 0 − U / Ω
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where ρ0 is the initial dislocation density (when ε=0). The relationship between flow stress and dislocation density is described as follows [28]:
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(2)
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σ = αµb ρ
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where α is a material constant, µ is the shear modulus, and b is the distance between
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atoms in the slip direction. Utilizing Eqs. (1) and (3), when dρ/dε=0, the dislocation in
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under steady-state conditions (ρsat) can be expressed as below:
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ρsat= U/Ω
(4)
Therefore, σsat can be expressed as follows:
σ sat = αµb U / Ω
(5)
Synthesizing Eqs. (2), (3) and (5), the flow stress during the work
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(3)
hardening-dynamic recovery periods can be described by the following equation:
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2 2 σ rec = σ sat + (σ 02 − σ sat ) e−Ωε
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where σrec is the flow stress during the work hardening-dynamic recovery periods
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(6)
and σ0 is the yield stress.
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The equations for the three parameters (σsat, σ0 and Ω) should be determined to
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predict the flow stress and the feature parameters of the true stress-strain curves are
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deeply influenced by the strain rate and temperature. The Zener-Hollomon parameter
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is widely used to characterize the relationship between the material feature parameters
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and the hot deformation parameters. The expression of the Zener-Hollomon parameter
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(Z) is given as below [29]:
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Z = ε& exp ( Q / RT )
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where ε& is the strain rate (s-1), R is the universal gas constant (8.314 J mol-1 K-1),
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T is the absolute temperature (K), and Q is the activation energy (kJ mol-1). σsat can be described by the following hyperbolic-sine type equation over a wide range of stress[5]:
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(7)
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ε& = A [sinh(ασ sat ) ] exp( −Q / RT )
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where A, α and n are constants that are independent of σsat and T. When the flow
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stress is low (ασ <0.8), Eq. (8) can be simplified according to an exponential law [30]
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to the following equation:
(8)
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n
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n ε& = Bσ sat
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When the flow stress is high (ασ>1.2), Eq. (8) can be simplified as follows [31]:
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ε& = C exp( βσ sat )
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Where B, C, β and n1 are material constants, and the value of α can be determined
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by β / n1 . The natural logarithm was taken on both sides of Eqs. (9) and (10) and the
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following equations can be respectively written as:
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ln(σ sat ) =
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1 1 ln(ε& ) − ln( B ) n1 n1
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(9)
(10)
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σ sat =
1
β
ln(ε& ) −
1
β
ln(C )
(11)
(12)
The relationship between ln σ sat − ln ε& and σ sat − ln ε& is given in Fig. 8. Through
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linear data fitting and averaging, the β and n1 values were calculated to be 9.205
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MPa-1 and 0.1346, respectively. The value of α was found to be 0.01462 MPa-1.
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To calculate the values of n, A and Q, by taking the natural logarithm of both sides
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of Eq. (8), the following equation can be expressed:
ln ε& Q ln A + − n ( nRT ) n
2
ln[sinh(ασ sat )] =
3
According to Eq. (13), the value of n can be obtained from the average value of the
(13)
slopes of ln [sinh(ασ sat )] − ln ε& at different temperatures as shown in Fig. 9, and the
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average value of n was calculated to be 6.80121. Then, Eq. (13) can also be given as
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follows:
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d {ln [sinh ασ sat ]}
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Q = Rn
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The mean value of activation energy Q was estimated to be 186 kJ/mol according
(14)
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to the linear relationship between ln [sinh(ασ sat )] and 1/T, as shown in Fig. 10. The
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calculated activation energy is close to the value of Q reported for an Al-Zn-Mg-Cu
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alloy in the literature [32].
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Finally, the value of lnA can been calculated according to the intercept of
ln [sinh(ασ sat )] − ln ε& when the value of Q is determined, and the value of A was
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calculated to be 6.68×1011 s-1.
Considering the definition of the hyperbolic law, the saturation stress at different
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hot working conditions can be approximately expressed as the function of the
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Zener-Hollomon parameter from Eqs. (7) and (8):
σ sat =
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1
α
sinh −1 (Z / A)1/ n
(15)
The yield stress (σ0) at various test conditions can be directly identified on the true
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stress-strain curves in terms of a 0.2% offset in the total strain. Fig. 11 shows the
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relationship between the yield stress and ln Z. Then, σ0 can be expressed as follows:
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σ 0 = 7.441ln Z − 141.661
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The coefficient of DRV (Ω) can be obtained through Eq. (6) for all the test
(16)
conditions. Fig. 12 shows the relationship between ln Ω and ln Z. Then, Ω can be
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expressed as a function of the Zener-Hollomon parameter:
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Ω = 9.12 × 10 2 Z −0.1635
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Consequently, the constitutive equations of the studied Al-Zn-Mg-Cu alloy during
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the work hardening-dynamic recovery periods under hot deformation can be presented
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as follows:
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σ = σ 2 + σ 2 − σ 2 e − Ωε 0.5 sat ) rec sat ( 0 −1 11 0.147 σ sat = 68.4sinh ( Z / 6.68 × 10 ) σ 0 = 7.441ln Z − 141.661 Ω = 9.12 × 102 Z −0.1635 Z = ε& exp(1.86 × 105 / RT )
(18)
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(17)
The Eq.(18) can also be used to predict the flow stress during the work hardening-dynamic recovery period for the true stress-strain curves controlled by
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DRX.
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3.5 Dynamic recrystallization modelling
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When the studied alloy is deformed at high temperatures or at low strain rates,
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DRX will be activated if the dislocation continually increases and accumulates to a
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threshold value after a certain level of deformation is reached (which can also be
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referred to the critical strain). The volume fraction of DRX (XD) can be described by
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the following expression [19]:
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nd ε − εc X D = 1 − exp −kd ε p
(ε ≥ ε c )
(19)
where kd and nd are the material constants. As shown in Fig. 6, the degree of DRX can be assumed as the difference (∆σ) between σdrx and σrec. Therefore, the volume
4
fraction of DRX can also be described as follows [33]: ∆σ σ − σ drx = rec ∆σ max σ sat − σ ss
(ε ≥ ε c )
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XD =
6
where σrec is the flow stress caused by work hardening and dynamic recovery
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during the DRX period and σdrx is the flow stress during the DRX period.
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(20)
The flow stress during the DRX period under hot deformation can be obtained by combining Eq. (19) with Eq. (20) as follows:
nd ε − ε c = σ rec − (σ sat − σ ss ) 1 − exp −kd ε p
(ε ≥ ε c )
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σ drx
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σss can be determined by the θ-σ curve as shown in Fig. 7 (b). Fig. 13 shows that a
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linear relation exists between the steady stress σss and ln Z, and σss can be represented
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as below:
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σ ss = 5.042 × ln Z − 87.415
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(21)
(22)
εp can be directly obtained by the measured flow stress-strain curves. Fig. 14 shows
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the relationship between ln(εp) and ln Z. Therefore, εp can be represented as a function
17
of Z :
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ε p = 0.0011× Z 0.1488
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Usually, it is difficult to determine accurately the quantitatively the values of εc, and
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0.8εp is used to replace εc according to previous studies [10, 34].
(23)
ACCEPTED MANUSCRIPT Then, substituting the true stress-strain data (after critical strain) into Eq. (21), the
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parameters kd and nd can be easily obtained for all the test conditions by a regression
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analysis. The calculated average value of nd is 2.2802. Fig. 15 shows that a linear
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relation exists between the material constants ln (kd) and ln (Z), and the material
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constant kd can be represented as a function of the Zener-Hollomon parameter (Z):
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kd = 0.0468 × Z 0.1124
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Therefore, the physically-based constitutive relation of the studied Al-Zn-Mg-Cu
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nd σ = σ − (σ − σ ) 1 − exp − k ε − ε c rec sat ss d drx εp 0.5 2 2 2 − Ωε σ rec = σ sat + (σ 0 − σ sat ) e 0.147 σ sat = 68.4 sinh −1 ( Z / 6.68 × 1011 ) σ 0 = 7.441ln Z − 141.661 Ω = 9.12 × 102 Z −0.1635 σ ss = 5.042 × ln Z − 87.41 ε = 0.8ε p c ε p = 0.0011 × Z 0.1488 kd = 0.0468 × Z 0.1124 nd = 1.5693 Z = ε& exp(1.86 × 105 / RT )
(ε ≥ ε c )
(25)
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3.6 Model validation
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alloy during the DRX period can be expressed as follows:
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(24)
The derived constitutive equations, Eqs. (18) and (25), can be used to predict the
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flow stress of the studied Al-Zn-Mg-Cu alloy over a wide range of strain values,
13
temperatures, and strain rates. To verify the accuracy of the constitutive equation,
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comparisons between the experimental and predicted results are analyzed as shown in
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Fig. 16 (a)-(e). All the predicted flow stresses show a good correlation with the
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measured flow stress values, whereas a small difference can be found in the flow
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stress at high strain when the temperature is relatively low. This finding is because an
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increased adiabatic temperature at larger strain values results in a reduction in the
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measured flow stress. The accuracy of the constitutive equation is further studied through calculations of
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the correlation coefficient (R) and the average absolute relative error (AARE). These
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parameters are defined below [35]:
∑ ( E − E ) ∑ ( P − P) 2
N
i =1
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( Ei − E )( Pi − P)
i
1 AARE ( % ) = N
i =1 N
N
∑ i =1
2
i
Pi − Ei × 100% Ei
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R=
N i =1
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∑
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(26)
(27)
where Ei and Pi are the measured and the predicted flow stress values, respectively,
E and P are the mean values of Ei and Pi, respectively, and N is the total number of
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data sets used in the study.
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The predicted values for the flow stress are plotted against the experimental results in Fig. 17. The correlation coefficient (R) and the average absolute relative error
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(AARE) values are 0.993 and 4.855%, respectively, which indicates a good predictive
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capability for the physically-based constitutive model to estimate the flow stress of
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the studied Al-Zn-Mg-Cu alloy.
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4. Conclusions In this study, the physically-based constitutive modelling of an Al-Zn-Mg-Cu alloy
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was carried out by isothermal hot compression tests over a wide range of temperatures
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(350~490 °C) and strain rates (0.001~1 s-1). Based on this investigation, important
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conclusions can be drawn: (1) During the hot compression process under all the above test conditions, the true
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stress-strain curves of the studied Al-Zn-Mg-Cu alloy can be divided into two distinct
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types. DRV is the main softening mechanism at high strain rates and low temperatures,
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and DRX is the predominant softening mechanism at low strain rates and high
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temperatures.
(2) The critical parameters under different hot compressed conditions were
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determined and the hot deformation activation energy of the studied Al-Zn-Mg-Cu
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alloy was estimated to be 186 kJ/mol under the test conditions.
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(3) The physically-based constitutive model is developed to predict the flow stress of the studied Al-Zn-Mg-Cu alloy during the hot compression tests. The correlation
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coefficient (R) and the average absolute relative error (AARE) values between the
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measured and predicted flow stresses are 0.993 and 4.855%, respectively. These
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results reveal that the physically-based constitutive model has a good prediction
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capability and can be used to determine the hot formation processing parameters for
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Al-Zn-Mg-Cu alloys.
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Acknowledgments
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The authors express their appreciation for the financial support of National Natural
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Science Foundation of China under Grant (No. 51405100), Shandong Provincial
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Natural Science Foundation, China, (No. ZR2017PA003), Postdoctoral Science
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Foundation of China under Grant (No. 2014M551233 and 2017T100237), Plan of
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Key Research and Development in Shandong Province under Grant (No.
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ACCEPTED MANUSCRIPT 2017GGX202006), and Plan of Co-Development of University in Weihai under Grant
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(No. 2016DXGJMS05).
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Reference
4
[1] X. Huang, H. Zhang, Y. Han, W. Wu, J. Chen, Hot deformation behavior of 2026
5
aluminum alloy during compression at elevated temperature, Mater. Sci. Eng. A 527
6
(2010) 485-490.
7
[2] Y.C. Lin, L.T. Li, Y.C. Xia, Y.Q. Jiang, Hot deformation and processing map of a
8
typical Al-Zn-Mg-Cu alloy, J. Alloys Comp. 550 (2013) 438-445.
9
[3] H. Mirzadeh, M. Roostaei, M.H. Parsa, R. Mahmudi, Rate controlling mechanisms
M AN U
SC
RI PT
1
during hot deformation of Mg-3Gd-1Zn magnesium alloy: Dislocation glide and
11
climb, dynamic recrystallization, and mechanical twinning, Mater. Des. 68 (2015)
12
228-231.
13
[4] D.X. Wen, Y.C. Lin, Y. Zhou, A new dynamic recrystallization kinetics model for a
14
Nb containing Ni-Fe-Cr-base superalloy considering influences of initial δ phase,
15
Vacuum 141 (2017) 316-327.
16
[5] C.M. Sellars, W.J. McTegart, On the mechanism of hot deformation, Acta Mater.
17
14 (1966) 1136-1138.
18
[6] Y.C. Lin, S.C. Luo, L.X. Yin, J. Huang, Microstructural evolution and high
19
temperature flow behaviors of a homogenized Sr-modified Al-Si-Mg alloy, J. Alloys
20
Comp. 739 (2017) 590-599.
21
[7] L.X. Li, Y. Lou, L.B. Yang, D.S. Peng, K.P. Rao, Flow stress behavior and
22
deformation characteristics of Ti-3Al-5V-5Mo compressed at elevated temperatures,
AC C
EP
TE D
10
ACCEPTED MANUSCRIPT Mater. Des. 23 (2002) 451-457.
2
[8] A. Momeni, H. Arabi, A. Rezaei, H. Badri, S.M. Abbasi, Hot deformation behavior
3
of austenite in HSLA-100 microalloyed steel, Mater. Sci. Eng. A 528 (2011)
4
2158-2163.
5
[9] N. Haghdadi, A. Zarei-Hanzaki, H.R. Abedi, The flow behavior modeling of cast
6
A356 aluminum alloy at elevated temperatures considering the effect of strain, Mater.
7
Sci. Eng. A 535 (2012) 252-257.
8
[10] Y.C. Lin, M.S. Chen, J. Zhong, Prediction of 42CrMo steel flow stress at high
9
temperature and strain rate, Mech. Res. Commun. 35 (2008) 142-150.
M AN U
SC
RI PT
1
[11] T. Yan, E. Yu, Y. Zhao, Constitutive modeling for flow stress of 55SiMnMo
11
bainite steel at hot working conditions, Mater. Des. 50 (2013) 574-580.
12
[12] Y.C. Lin, X.M. Chen, D.X. Wen, M.S. Chen, A physically-based constitutive
13
model for a typical nickel-based superalloy, Comput. Mater. Sci. 83 (2014) 282-289.
14
[13] G. Ji, Q. Li, L. Li, A physically-based constitutive relation to predict flow stress
15
for Cu-0.4Mg alloy during hot working, Mater. Sci. Eng. A 615 (2014) 247-254.
16
[14] P.M. Souza, H. Beladi, R. Singh, B. Rolfe, P.D. Hodgson, Constitutive analysis
17
of hot deformation behavior of a Ti6Al4V alloy using physical based model, Mater.
18
Sci. Eng. A 648 (2015) 265-273.
19
[15] N. Hghdadi, D. Martin, P. Hodgson, Physically-based constitutive modelling of
20
hot deformation behavior in a LDX 2101 duplex stainless steel, Mater. Des. 106 (2016)
21
420-427.
22
[16] R. Zhu, Q. Liu, J. Li, S. Xiang, Y. Chen, X. Zhang, Dynamic restoration
AC C
EP
TE D
10
ACCEPTED MANUSCRIPT mechanism and physically based constitutive model of 2050 Al-Li alloy during hot
2
compression, J. Alloys Comp. 650 (2015) 75-85.
3
[17] J. Zuo, L. Hou, J. Shi, H. Cui, L. Zhuang, J. Zhang, Effect of deformation
4
induced precipitation on grain refinement and improvement of mechanical properties
5
AA 7055 aluminum alloy, Mater. Charact. 130 (2017) 123-134.
6
[18] G. Chen, F.Y. Lin, S.J. Yao, F. Han, B. Wei, Y.M. Zhang,
7
of aluminum alloy in a wide temperature range from warm to semi-solid regions, J.
8
Alloys Comp. 674 (2016) 26-36.
9
[19] B. Wu, M.Q. Li, D.W. Ma, The flow behavior and constitutive equations in
10
isothermal compression of 7050 aluminum alloy, Mater. Sci. Eng. A 542 (2012)
11
79-87.
12
[20] G. Meng, B.L. Li, H.M. Li, H. Huang, Z.R. Nie, Hot deformation and processing
13
maps of an Al-5.7 wt.% Mg alloy with erbium, Mater. Sci. Eng. A 517 (2009)
14
132-137.
15
[21] X.Y. Liu, Q.L. Pan, Y.B. He, W.B. Li, W.J. Liang, Z.M. Yin, Flow behavior and
16
microstructural evolution of Al-Cu-Mg-Ag alloy during hot compression deformation,
17
Mater. Sci. Eng. A 500 (2009) 150-154.
18
[22] S. Gourdet, F. Montheillet, A model of continuous dynamic recrystallization,
19
Acta Mater. 51 (2003) 2685-2699.
20
[23] D.D. Chen, Y.C. Lin, Y. Zhou, M.S. Chen, D.X. Wen, Dislocation substructures
21
evolution and an adaptive-network-based fuzzy inference system model for
22
constitutive behavior of a Ni-based superalloy during hot deformation, J. Alloys
RI PT
1
AC C
EP
TE D
M AN U
SC
Constitutive behavior
ACCEPTED MANUSCRIPT Comp. 708 (2017) 938-946.
2
[24] N. Haghdadi, P. Cizek, H. Beladi, P.D. Hodgson. A novel high-strain-rate ferrite
3
dynamic softening mechanism facilitated by the interphase in the austenite/ferrite
4
microstructure, Acta Mater. 126 (2017) 44-57.
5
[25] M.R. Rokni, A. Zarei-Hanzaki, A.A. Roostaei, A. Abolhasani, Constitutive base
6
analysis of a 7075 aluminum alloy during hot compression testing, Mater. Des. 32
7
(2011) 4955-4960.
8
[26] N. Jin, H. Zhang, Y. Han, W. Wu, J. Chen, Hot deformation behavior of 7150
9
aluminum alloy during compression at elevated temperature, Mater. Charact. 60 (2009)
M AN U
SC
RI PT
1
530-536.
11
[27] E. Cerri, E. Evangelista, A. Forcellese, H.J. McQueen, Comparative hot
12
workability of 7012 and 7075 alloys after different pretreatments, Mater. Sci. Eng. A
13
197 (1995) 181-198.
14
[28] E.P. Busso, F.A. Mcclintock, A dislocation mechanics-based crystallographic
15
model of a B2-type intermetallic alloy, Int. J. Plast. 12 (1996) 1-28.
16
[29] C. Zener, J.H.Hollomon, Effect of strain rate upon plastic flow of steel, J. Appl.
17
Phys. 15 (1944) 22-32.
18
[30] J. Jonas, C.M. Sellars, W.J. Tegart, Strength and structure under hot-working
19
conditions, Int. Mater. Rev. 14 (1969) 1-24.
20
[31] H. Shi, A.J. Mclaren, C.M. Sellars, R. Shahani, R. Bolingbroke, Constitutive
21
equations for high temperature flow stress of aluminium alloys, Mater. Sci. Technol.
22
13 (1997) 210-216.
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TE D
10
ACCEPTED MANUSCRIPT [32] M. Zhou, Y.C. Lin, J. Deng, Y.Q. Jiang, Hot tensile deformation behaviors and
2
constitutive model of an Al-Zn-Mg-Cu alloy, Mater. Des. 59 (2014) 141-150.
3
[33] A. Laasraoui, J.J. Jonas, Prediction of steel flow stresses at high temperatures and
4
strain rates, Metall. Mater. Trans. A 22 (1991) 1545-1598.
5
[34] X.M. Chen, Y.C. Lin, D.X. Wen, J.L. Zhang, M. He, Dynamic recrystallization
6
behavior of a typical nickel-based superalloy during hot deformation, Mater. Des. 57
7
(2014) 568-577.
8
[35] N. Haghdadi, A. Zarei-Hanzaki, A.R. Khalesian, H.R. Abedi. Artificial neural
9
network modeling to predict the hot deformation behavior of an A356 aluminum alloy,
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Mater. Des. 49 (2013) 386-391.
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Figure Captions
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Fig. 1 Initial microstructure of the extruded Al-Mg-Zn-Cu alloy sample.
Fig. 2 Microstructures of the studied Al-Mg-Zn-Cu alloy deformed at (a) 350 °C, (b)
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400 °C, (c) 450 °C, and (d) 470 °C with a strain rate of 0.01 s-1.
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Fig. 3 Microstructures of the studied Al-Mg-Zn-Cu alloy deformed at 470 °C with
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strain rates of (a) 1 s-1, (b) 0.1 s-1, (c) 0.01 s-1, and (d) 0.001 s-1.
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Fig. 4 TEM images of the studied Al-Mg-Zn-Cu alloy deformed at (a) 350 °C/0.001
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s-1, (b) 470 °C/0.1 s-1, and (c) 470 °C /0.001 s-1.
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Fig. 5 True strain-stress curves at different strain rates: (a) 0.001 s-1, (b) 0.01 s-1, (c)
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0.1 s-1, and (d) 1 s-1.
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Fig. 6 A schematic diagram of true stress-strain curves (Symbols ‘a’ and ‘b’ show the
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dynamic recovery and dynamic recrystallization mechanisms, respectively.).
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Fig. 7 (a) Schematic of the θ-σ curves for the studied Al-Mg-Zn-Cu alloy at various
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temperatures and strain rates. (b) The θ-σ curve at a temperature 470 °C with a strain
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rate of 0.001 s-1.
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Fig. 9 Linear relationship fitting of ln(sinh(ασsat))- ln ε& .
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Fig. 8 The relationships between (a) lnσsat and ln ε& , and (b) σsat and ln ε& .
Fig. 10 Linear relationship fitting of ln(sinh(ασsat))-1/T.
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Fig. 11 Relationship between the yield stress (σ0) and the Zener-Hollomon parameter.
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Fig. 12 Relationship between the coefficient of dynamic recovery (Ω) and the
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Zener-Hollomon parameter.
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Fig. 13 Relationship between the steady stress (σss) and the Zener-Hollomon
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parameter.
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Fig. 14 Relationship between the peak strain (εp) and the Zener-Hollomon parameter.
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Fig. 15 Relationship between the materials constants (kd) and the Zener-Hollomon
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parameter.
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Fig. 16 Comparisons between the predicted and measured flow stresses of the studied
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Al-Mg-Zn-Cu alloy at temperatures of (a) 350 °C, (b) 400 °C, (c) 450 °C, (d) 470 °C,
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and (e) 490 °C.
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Fig. 17 Correlation between the predicted and experimental flow stress data.
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Fig. 1 Initial microstructure of the extruded Al-Mg-Zn-Cu alloy.
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Fig. 2 Microstructures of the studied Al-Mg-Zn-Cu alloy deformed at (a) 350 °C, (b)
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Fig. 3 Microstructures of the studied Al-Mg-Zn-Cu alloy deformed at 470 °C with
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Fig. 4 TEM images of the studied Al-Mg-Zn-Cu alloy deformed at (a) 350 °C/0.001
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Fig. 5 True strain-stress curves at different strain rates: (a) 0.001 s-1, (b) 0.01 s-1, (c)
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Fig. 6 A schematic diagram of true stress-strain curves (Symbols ‘a’ and ‘b’ show the
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Fig. 7 (a) Schematic of the θ-σ curves for the studied Al-Mg-Zn-Cu alloy at various temperatures and strain rates. (b) The θ-σ curve at a temperature 470 °C with a strain
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Fig. 9 Linear relationship fitting of ln(sinh(ασsat))- ln ε& .
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Fig. 11 Relationship between the yield stress (σ0) and the Zener-Hollomon parameter.
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Fig. 16 Comparisons between the predicted and measured flow stresses of the studied Al-Mg-Zn-Cu alloy at temperatures of (a) 350 °C, (b) 400 °C, (c) 450 °C, (d) 470 °C, and (e) 490 °C.
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Fig. 17 Correlation between the predicted and experimental flow stress data.
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The hot deformation behavior of an Al-Zn-Mg-Cu alloy was studied by isothermal hot compression tests.
The hot deformation activation energy of the Al-Zn-Mg-Cu alloy was estimated to be 186 kJ/mol.
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The physically-based constitutive model of the Al-Zn-Mg-Cu alloy was
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DOI:
10.1016/j.jallcom.2018.02.039
Reference:
JALCOM 44915
To appear in:
Journal of Alloys and Compounds
Received Date: 30 October 2017 Revised Date:
4 February 2018
Accepted Date: 5 February 2018
Please cite this article as: H. Zhang, G. Chen, Q. Chen, F. Han, Z. Zhao, A physically-based constitutive modelling of a high strength aluminum alloy at hot working conditions, Journal of Alloys and Compounds (2018), doi: 10.1016/j.jallcom.2018.02.039. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A physically-based constitutive modelling of a high strength aluminum alloy at
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hot working conditions
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Hongming Zhang a, Gang Chen b,*, Qiang Chen c,*, Fei Han b, Zude Zhao c
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a
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China
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b
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Weihai 264209, China
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c
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China
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Department of Civil Engineering, Harbin Institute of Technology, Weihai 264209,
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School of Materials Science and Engineering, Harbin Institute of Technology,
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Southwest Technology and Engineering Research Institute, Chongqing 400039,
* Corresponding author. E-mail address: [email protected] (Gang Chen); [email protected] (Qiang Chen)
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Abstract: The hot deformation behavior of a high strength aluminum alloy
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(Al-Zn-Mg-Cu) was studied by isothermal hot compression tests performed over a
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range of temperatures (350~490 °C) and strain rates (0.001~1 s-1). A constitutive
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equation was established using experimental results to predict the flow stress of the
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alloy under elevated temperature. In the work hardening-dynamic recovery regime, a
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physically-based constitutive equation for the flow stress was obtained from the
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stress-dislocation relation. In the subsequent dynamic recrystallization region, the
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flow stress after the peak was predicted by employing the kinematics of the dynamic
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recrystallization in the constitutive model. The stress-strain curves of the alloy
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predicted by the established models were in good agreement with experimental results.
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The results indicate that the proposed physically-based constitutive equation can
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accurately predict the flow behavior of the Al-Zn-Mg-Cu alloy.
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Keywords: Al-Zn-Mg-Cu alloy; Hot deformation behavior; Dynamic recovery and
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dynamic recrystallization; Physically-based constitutive model
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1. Introduction Al-Zn-Mg-Cu alloys belong to the high strength aluminum alloys, which have been
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widely used to manufacture airplane structures and auto parts due to high
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strength-to-weight ratio and good fatigue resistance properties. Due to lower plasticity
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of Al-Zn-Mg-Cu alloys at room temperature, hot plastic forming is usually chosen to
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process the materials. During the hot deformation process, the microstructures of the
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material change significantly because of the influence of temperatures, strains and
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strain rates [1, 2]. Furthermore, the microstructural evolution strongly affects the flow
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stress behavior [3, 4]. To design the thermomechanical processing parameters for
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Al-Zn-Mg-Cu alloys, it is necessary to study their hot deformation behavior.
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The constitutive equations, which establish correlations with the flow stress, strain rate and temperature, are often used to estimate the flow behavior of metal materials
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at hot working conditions. The stress-strain curves obtained by the uniaxial hot
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compression tests are often employed to provide the necessary data for the
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constitutive equations. In the previous investigations, various models have been
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proposed to predict the flow behavior of metal materials. Sellars and Tegart proposed
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the phenomenological approach in which the flow stress was described by the
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hyperbolic-sine Arrhenius equation [5], and much work has been done to study the
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flow behavior of various metal materials by the phenomenological model based on the
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hyperbolic-sine law [6-9]. The phenomenological model is rather straightforward,
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However this method does not fully consider the microstructural effects, and lacks
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physical meaning. In the plastic deformation, the dislocation density variation is one
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of the most important microstructural parameters. Based on the dislocation density
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variation, the physical constitutive model can be built which correlates with the
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physical mechanism of the work hardening (WH), dynamic recovery (DRV) and
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dynamic recrystallization (DRX). A number of physically-based constitutive models
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have been established to describe the hot deformation behavior of metal materials,
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such as 42CrMo steel [10], 55SiMnMo bainite steel [11], nickel-based super-alloy
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[12], Cu-Mg alloy [13], Ti6Al4V alloy [14], LDX 2101 duplex stainless steel [15],
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and 2050 Al-Li alloy [16]. These studies make accurate predictions for the
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experimental results, indicating that the physically-based constitutive models can
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accurately describe the flow stress of the metal materials at hot deformation
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conditions. Generally, an Al-Zn-Mg-Cu alloy has high stacking fault energy, and the
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insoluble dispersed phases exist in the alloy [17]. These factors seriously affect the
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dynamic recrystallization behavior during the hot deformation of Al-Zn-Mg-Cu alloys.
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Therefore, it is necessary to develop an accurate physically-based constitutive model
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to properly account for the intricate effect of hot working conditions on the flow
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behavior of Al-Mg-Zn-Cu alloys.
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The objective of the present work is to investigate the hot deformation
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characteristics of an Al-Zn-Mg-Cu alloy by utilizing the uniaxial hot compression
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tests and to establish a physically-based constitutive model to describe the flow stress
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of the alloy during the work hardening-dynamic recovery and dynamical
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recrystallization periods. Then the predictive performance of the developed
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constitutive model is discussed by comparing with the experimental data.
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2.
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Experimental procedure A high-strength aluminum alloy with a composition of Al-5.65Zn-2.18Mg-1.71Cu
(wt. %) was adopted for this work. The studied Al-Zn-Mg-Cu alloy was first extruded
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at 400 °C with an extrusion ratio of 17 and then machined to compression specimens
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with diameters of 8 mm and heights of 12 mm.
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According to the previous study by the authors regarding the plastic deformation characteristics of Al-Zn-Mg-Cu alloys in a warm deformation field [18], the hot
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compression tests were carried out in a temperature range of 350~490 °C and a strain
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rate range of 0.001~1 s-1 on a Gleeble 1500 thermomechanical simulator, which can
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automatically record true stress-true strain data. To minimize the friction between the
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specimens and the die during hot deformation, the flat ends of the specimen were
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recessed to entrap the high-temperature graphite lubricant. The thermocouple wires
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were welded onto the surface of the hot compressed specimen to measure the
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temperature of the specimens. The specimens were heated to a preset temperature at a
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rate of 10 °C/s, soaked for 3 min to ensure a uniform temperature over the entire
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specimen. After each compression test, the deformed specimens were quenched in
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water quickly to obtain the hot deformation microstructures. The microstructures of
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hot compressed specimen were examined by the Olympus PMG3 optical microscope
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and a JEOL-2100 transmission electron microscope.
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Results and discussion
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3.1 Microstructural evolution
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Fig. 1 shows the initial microstructure of the extruded Al-Mg-Zn-Cu alloy. The results indicate the grains were fibrous and DRX did not occur in the microstructures.
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Moreover, the second phases were lath-like and distributed along the extrusion
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direction.
Fig. 2 shows the microstructures of the studied Al-Zn-Mg-Cu alloy compressed at a
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strain rate of 0.01 s-1 under different temperatures. As shown in Fig. 2 (a) and (b), the
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equiaxial recrystallized microstructures were not found after compression at 350 °C
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and 400 °C. When the deformation temperature was increased to 450 °C, some fine
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recrystallized grains were detected in the microstructures (Fig. 2(c)), indicating DRX
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had occurred during deformation. As the deformation temperature increased, a large
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amount of recrystallized grains appeared at 470 °C (Fig. 2(d)). It is assumed that
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higher temperatures can activate DRX and alter the softening mechanism [19].
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Fig. 3 shows the microstructures of the studied Al-Zn-Mg-Cu alloy compressed at
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8
470 °C under different strain rates. Fig. 3(a) exhibits the microstructures at a higher
17
strain rate of 1 s-1, and the results demonstrate that only a few microstructures had
18
been recrystallized. When the alloy was compressed at lower strain rates (0.1 s-1 and
19
0.01 s-1), the recrystallized microstructures increased gradually. It is assumed that
20
DRX is the predominant softening mechanism, whereas the softening caused by DRV
21
is weak (Fig. 3(b) and (c)). When the alloy was deformed at a strain rate of 0.001 s-1,
22
the microstructures consisted of subgrains and fine recrystallized grains, indicating
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16
ACCEPTED MANUSCRIPT that full DRX was attained during deformation (Fig. 3(d)). Therefore, the strain rate
2
plays an important role in DRV and DRX during hot deformation. At lower strain
3
rates, the dislocations have sufficient time to climb or slip into subgrain boundaries,
4
and when the low-angle grain boundaries have transformed to large-angle grain
5
boundaries, most of the subgrains can evolve into recrystallized grains [20].
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Fig. 4 shows the TEM micrographs of the studied Al-Zn-Mg-Cu alloy compressed
7
at 350 °C /0.001 s-1, 470 °C /0.1 s-1 and 470 °C /0.001 s-1. As shown in Fig. 4(a), when
8
the alloy was compressed at 350 °C/0.001 s-1, the dislocation underwent a multilateral
9
movement based on the dislocation climb and slip, which occurred during the DRV
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6
stage and effectively reduced the dislocation density [21]. When the alloy was
11
compressed at 470 °C/0.1 s-1, dislocation dissociation was detected, indicating that the
12
nucleation caused by subgrain merging occurred and DRX participated in the hot
13
deformation (Fig. 4(b)) [2]. When the alloy was compressed at 470 °C/0.001 s-1, the
14
grains with typical DRX characteristics can be determined according to the clean and
15
straight high-angle grain boundaries (Fig. 4(c)) [22,23]. The specifics of the
16
microstructures in Fig. 4 suggest a conclusion that DRX effect works at higher
17
temperatures and with lower strain rates.
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Haghdadi et al. recently reported a distinct softening mechanism analogous to
19
discontinuous dynamic recrystallization within ferrite at a high strain rate [24].
20
However, the stacking fault energy of an aluminum alloy is quite high resulting in
21
recrystallization difficulties [17], and recrystallization time is insufficient at high
22
strain rates. Therefore, the opposite tendency appears compared with that described in
ACCEPTED MANUSCRIPT 1
the report by Haghdadi et al.
2
3.2 Hot deformation behavior
3
The true stress-strain curves of the studied Al-Zn-Mg-Cu alloy during hot compression are shown in Fig. 5. It can be found that the peak stress increases as the
5
deformation temperature decreases and the strain rate increases. This finding is
6
because the higher strain rate causes more tangled dislocation structures, which hinder
7
the dislocation movement and increase the resistance of the material deformation [9
8
25]. Moreover, lower temperatures slow down thermally activated processes, resulting
9
in insufficient motion of the dislocations and vacancies [19]. Furthermore, higher
M AN U
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4
strain rates can reduce the coarsening time of dynamically recrystallized grains, and
11
the lower temperatures inhibit the mobility of the recrystallized grain boundaries,
12
which also lead to increasing the flow stress [26].
13
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10
As indicated in Fig. 5, the stress-strain curves can be divided into two distinct types, and the schematic curves marked with ‘a’ and ‘b’ in Fig. 6 indicate the different
15
characteristics between two types of stress-strain curves. The characteristic of the
16
stress curve marked with ‘a’ is attributed to DRV as the main softening mechanism. At
17
the beginning of the deformation, the flow stress rapidly increases with an increase in
18
strain due to the work hardening. When a balance between work hardening and
19
dynamic recovery is reached, a saturation value (σsat) appears and remains constant.
20
For the curve marked with ‘b’, it is considered that DRX is the dominant softening
21
mechanism. The stress-strain curve marked with ‘b’ consists of three stages: stage I
22
(work hardening stage), stage II (softening stage) and stage III (steady stage). In stage
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ACCEPTED MANUSCRIPT I, the flow stress rapidly increases to a critical stress (σc) at the initial deformation
2
stage due to the work hardening. Once the deformation value exceeds the critical
3
strain (εc), DRX occurs and the flow stress moves into stage II. It can be found that
4
the flow stress continues to slowly increase to the peak stress (σp) because of the
5
contest between the work hardening and dynamic softening mechanisms (including
6
DRV and DRX). When the deformation exceeds the peak strain (εp), the dynamic
7
softening caused by DRX becomes strong enough to overcome the work hardening,
8
then the flow stress begins to decrease until it reaches a steady stress (σss). During
9
stage III, a balance between work hardening and dynamic softening is reached, and
10
the flow stress remains unchanged regardless of the increased strain. In Fig. 6, σrec
11
represents the saturation stress during the steady-state deformation due to DRV, and
12
σdrx represents the steady-state stress due to DRX. Assuming that DRX does not
13
happen in the curve marked with ‘a’, the difference (∆σ) between σdrx and σrec is the
14
net softening attributable to DRX.
15
3.3 Work hardening behavior
16
It is important to analyze the work hardening behavior of the alloy, which can
17
clearly reflect the evolution in DRV. The relation between the strain-hardening rate
18
(θ=dσ/dε) and the flow stress has been widely used to represent the work hardening
19
behavior of alloys. Fig. 7 shows that the θ-σ curve of the studied Al-Zn-Mg-Cu alloy
20
is compressed at different temperatures and strain rates. As shown in Fig. 7 (a), the
21
work hardening rate decreases as the strain rate decreases and the temperature
22
increases. During plastic deformation, the work hardening rate depends on the
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ACCEPTED MANUSCRIPT 1
competition between the dislocation storage and annihilation, and the effect of DRV is
2
enhanced as the strain rate decreases and the temperature increases.
3
The fitting θ-σ curve at 470 °C/0.001 s-1 is shown in Fig. 7 (b). Initially, the value of θ decreases with an increase in stress, due to the softening effect induced by DRV.
5
Then, an inflection appears in the θ-σ curve, which indicates that DRX occurs under
6
the deformation conditions. Moreover, the inflection point in the θ-σ curve is
7
generally used to determine the value of σc [13]. When the strain-hardening rate θ
8
becomes 0, σp is achieved. The stress continuously decreases as the strain increases
9
until the stress reaches σss, when the value θ becomes zero again. The horizontal
10
intercept of the tangent line of the θ-σ curve through the inflection point can be
11
regarded as σsat.
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3.4 Dynamic recovery modelling
13
The variation in flow stress during hot deformation, which is under the control of
14
work hardening and dynamic recovery is primarily attributed to the evolution in the
15
dislocation density with strain. It is generally considered that the work hardening
16
increases the number of dislocations and the dynamic recovery gives rise to
17
dislocation annihilation and rearrangement. The evolution in the dislocation density
18
with strain can be expressed as the following two components [27]:
19
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12
20
where ρ is the dislocation density, ε is the strain, dρ/dε is the increasing rate of the
d ρ / d ε = U − Ωρ
(1)
21
dislocation density with strain; and U and Ω are the parameters with respect to the
22
strain that represent the work hardening and softening terms, respectively. The
ACCEPTED MANUSCRIPT 1
dislocation density can be defined by integrating Eq. (1) as follows:
2
ρ = e − Ωε U / Ω e Ωε + ρ 0 − U / Ω
3
where ρ0 is the initial dislocation density (when ε=0). The relationship between flow stress and dislocation density is described as follows [28]:
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4
(2)
5
σ = αµb ρ
6
where α is a material constant, µ is the shear modulus, and b is the distance between
7
atoms in the slip direction. Utilizing Eqs. (1) and (3), when dρ/dε=0, the dislocation in
8
under steady-state conditions (ρsat) can be expressed as below:
11 12 13
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10
ρsat= U/Ω
(4)
Therefore, σsat can be expressed as follows:
σ sat = αµb U / Ω
(5)
Synthesizing Eqs. (2), (3) and (5), the flow stress during the work
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9
(3)
hardening-dynamic recovery periods can be described by the following equation:
14
2 2 σ rec = σ sat + (σ 02 − σ sat ) e−Ωε
15
where σrec is the flow stress during the work hardening-dynamic recovery periods
17
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(6)
and σ0 is the yield stress.
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0.5
The equations for the three parameters (σsat, σ0 and Ω) should be determined to
18
predict the flow stress and the feature parameters of the true stress-strain curves are
19
deeply influenced by the strain rate and temperature. The Zener-Hollomon parameter
20
is widely used to characterize the relationship between the material feature parameters
21
and the hot deformation parameters. The expression of the Zener-Hollomon parameter
22
(Z) is given as below [29]:
ACCEPTED MANUSCRIPT 1
Z = ε& exp ( Q / RT )
2
where ε& is the strain rate (s-1), R is the universal gas constant (8.314 J mol-1 K-1),
4 5
T is the absolute temperature (K), and Q is the activation energy (kJ mol-1). σsat can be described by the following hyperbolic-sine type equation over a wide range of stress[5]:
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(7)
6
ε& = A [sinh(ασ sat ) ] exp( −Q / RT )
7
where A, α and n are constants that are independent of σsat and T. When the flow
8
stress is low (ασ <0.8), Eq. (8) can be simplified according to an exponential law [30]
9
to the following equation:
(8)
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n
10
n ε& = Bσ sat
11
When the flow stress is high (ασ>1.2), Eq. (8) can be simplified as follows [31]:
12
ε& = C exp( βσ sat )
13
Where B, C, β and n1 are material constants, and the value of α can be determined
14
by β / n1 . The natural logarithm was taken on both sides of Eqs. (9) and (10) and the
15
following equations can be respectively written as:
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ln(σ sat ) =
17
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1 1 ln(ε& ) − ln( B ) n1 n1
16
18
(9)
(10)
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1
σ sat =
1
β
ln(ε& ) −
1
β
ln(C )
(11)
(12)
The relationship between ln σ sat − ln ε& and σ sat − ln ε& is given in Fig. 8. Through
19
linear data fitting and averaging, the β and n1 values were calculated to be 9.205
20
MPa-1 and 0.1346, respectively. The value of α was found to be 0.01462 MPa-1.
21
To calculate the values of n, A and Q, by taking the natural logarithm of both sides
ACCEPTED MANUSCRIPT 1
of Eq. (8), the following equation can be expressed:
ln ε& Q ln A + − n ( nRT ) n
2
ln[sinh(ασ sat )] =
3
According to Eq. (13), the value of n can be obtained from the average value of the
(13)
slopes of ln [sinh(ασ sat )] − ln ε& at different temperatures as shown in Fig. 9, and the
5
average value of n was calculated to be 6.80121. Then, Eq. (13) can also be given as
6
follows:
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d {ln [sinh ασ sat ]}
7
Q = Rn
8
The mean value of activation energy Q was estimated to be 186 kJ/mol according
(14)
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d (1 / T )
to the linear relationship between ln [sinh(ασ sat )] and 1/T, as shown in Fig. 10. The
10
calculated activation energy is close to the value of Q reported for an Al-Zn-Mg-Cu
11
alloy in the literature [32].
12
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9
Finally, the value of lnA can been calculated according to the intercept of
ln [sinh(ασ sat )] − ln ε& when the value of Q is determined, and the value of A was
14
calculated to be 6.68×1011 s-1.
Considering the definition of the hyperbolic law, the saturation stress at different
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13
16
hot working conditions can be approximately expressed as the function of the
17
Zener-Hollomon parameter from Eqs. (7) and (8):
σ sat =
18 19
1
α
sinh −1 (Z / A)1/ n
(15)
The yield stress (σ0) at various test conditions can be directly identified on the true
20
stress-strain curves in terms of a 0.2% offset in the total strain. Fig. 11 shows the
21
relationship between the yield stress and ln Z. Then, σ0 can be expressed as follows:
ACCEPTED MANUSCRIPT 1
σ 0 = 7.441ln Z − 141.661
2
The coefficient of DRV (Ω) can be obtained through Eq. (6) for all the test
(16)
conditions. Fig. 12 shows the relationship between ln Ω and ln Z. Then, Ω can be
4
expressed as a function of the Zener-Hollomon parameter:
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5
Ω = 9.12 × 10 2 Z −0.1635
6
Consequently, the constitutive equations of the studied Al-Zn-Mg-Cu alloy during
7
the work hardening-dynamic recovery periods under hot deformation can be presented
8
as follows:
10
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σ = σ 2 + σ 2 − σ 2 e − Ωε 0.5 sat ) rec sat ( 0 −1 11 0.147 σ sat = 68.4sinh ( Z / 6.68 × 10 ) σ 0 = 7.441ln Z − 141.661 Ω = 9.12 × 102 Z −0.1635 Z = ε& exp(1.86 × 105 / RT )
(18)
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(17)
The Eq.(18) can also be used to predict the flow stress during the work hardening-dynamic recovery period for the true stress-strain curves controlled by
12
DRX.
14
3.5 Dynamic recrystallization modelling
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When the studied alloy is deformed at high temperatures or at low strain rates,
15
DRX will be activated if the dislocation continually increases and accumulates to a
16
threshold value after a certain level of deformation is reached (which can also be
17
referred to the critical strain). The volume fraction of DRX (XD) can be described by
18
the following expression [19]:
ACCEPTED MANUSCRIPT 1
2
nd ε − εc X D = 1 − exp −kd ε p
(ε ≥ ε c )
(19)
where kd and nd are the material constants. As shown in Fig. 6, the degree of DRX can be assumed as the difference (∆σ) between σdrx and σrec. Therefore, the volume
4
fraction of DRX can also be described as follows [33]: ∆σ σ − σ drx = rec ∆σ max σ sat − σ ss
(ε ≥ ε c )
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3
5
XD =
6
where σrec is the flow stress caused by work hardening and dynamic recovery
8 9
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during the DRX period and σdrx is the flow stress during the DRX period.
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7
(20)
The flow stress during the DRX period under hot deformation can be obtained by combining Eq. (19) with Eq. (20) as follows:
nd ε − ε c = σ rec − (σ sat − σ ss ) 1 − exp −kd ε p
(ε ≥ ε c )
10
σ drx
11
σss can be determined by the θ-σ curve as shown in Fig. 7 (b). Fig. 13 shows that a
12
linear relation exists between the steady stress σss and ln Z, and σss can be represented
13
as below:
15
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σ ss = 5.042 × ln Z − 87.415
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14
(21)
(22)
εp can be directly obtained by the measured flow stress-strain curves. Fig. 14 shows
16
the relationship between ln(εp) and ln Z. Therefore, εp can be represented as a function
17
of Z :
18
ε p = 0.0011× Z 0.1488
19
Usually, it is difficult to determine accurately the quantitatively the values of εc, and
20
0.8εp is used to replace εc according to previous studies [10, 34].
(23)
ACCEPTED MANUSCRIPT Then, substituting the true stress-strain data (after critical strain) into Eq. (21), the
2
parameters kd and nd can be easily obtained for all the test conditions by a regression
3
analysis. The calculated average value of nd is 2.2802. Fig. 15 shows that a linear
4
relation exists between the material constants ln (kd) and ln (Z), and the material
5
constant kd can be represented as a function of the Zener-Hollomon parameter (Z):
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1
6
kd = 0.0468 × Z 0.1124
7
Therefore, the physically-based constitutive relation of the studied Al-Zn-Mg-Cu
SC
nd σ = σ − (σ − σ ) 1 − exp − k ε − ε c rec sat ss d drx εp 0.5 2 2 2 − Ωε σ rec = σ sat + (σ 0 − σ sat ) e 0.147 σ sat = 68.4 sinh −1 ( Z / 6.68 × 1011 ) σ 0 = 7.441ln Z − 141.661 Ω = 9.12 × 102 Z −0.1635 σ ss = 5.042 × ln Z − 87.41 ε = 0.8ε p c ε p = 0.0011 × Z 0.1488 kd = 0.0468 × Z 0.1124 nd = 1.5693 Z = ε& exp(1.86 × 105 / RT )
(ε ≥ ε c )
(25)
11
3.6 Model validation
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10
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9
alloy during the DRX period can be expressed as follows:
M AN U
8
(24)
The derived constitutive equations, Eqs. (18) and (25), can be used to predict the
12
flow stress of the studied Al-Zn-Mg-Cu alloy over a wide range of strain values,
13
temperatures, and strain rates. To verify the accuracy of the constitutive equation,
14
comparisons between the experimental and predicted results are analyzed as shown in
15
Fig. 16 (a)-(e). All the predicted flow stresses show a good correlation with the
16
measured flow stress values, whereas a small difference can be found in the flow
ACCEPTED MANUSCRIPT 1
stress at high strain when the temperature is relatively low. This finding is because an
2
increased adiabatic temperature at larger strain values results in a reduction in the
3
measured flow stress. The accuracy of the constitutive equation is further studied through calculations of
5
the correlation coefficient (R) and the average absolute relative error (AARE). These
6
parameters are defined below [35]:
∑ ( E − E ) ∑ ( P − P) 2
N
i =1
8
9
( Ei − E )( Pi − P)
i
1 AARE ( % ) = N
i =1 N
N
∑ i =1
2
i
Pi − Ei × 100% Ei
SC
R=
N i =1
M AN U
7
∑
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4
(26)
(27)
where Ei and Pi are the measured and the predicted flow stress values, respectively,
E and P are the mean values of Ei and Pi, respectively, and N is the total number of
11
data sets used in the study.
12
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10
The predicted values for the flow stress are plotted against the experimental results in Fig. 17. The correlation coefficient (R) and the average absolute relative error
14
(AARE) values are 0.993 and 4.855%, respectively, which indicates a good predictive
15
capability for the physically-based constitutive model to estimate the flow stress of
16
the studied Al-Zn-Mg-Cu alloy.
18 19
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4. Conclusions In this study, the physically-based constitutive modelling of an Al-Zn-Mg-Cu alloy
20
was carried out by isothermal hot compression tests over a wide range of temperatures
21
(350~490 °C) and strain rates (0.001~1 s-1). Based on this investigation, important
ACCEPTED MANUSCRIPT 1
conclusions can be drawn: (1) During the hot compression process under all the above test conditions, the true
3
stress-strain curves of the studied Al-Zn-Mg-Cu alloy can be divided into two distinct
4
types. DRV is the main softening mechanism at high strain rates and low temperatures,
5
and DRX is the predominant softening mechanism at low strain rates and high
6
temperatures.
(2) The critical parameters under different hot compressed conditions were
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7
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2
determined and the hot deformation activation energy of the studied Al-Zn-Mg-Cu
9
alloy was estimated to be 186 kJ/mol under the test conditions.
10
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8
(3) The physically-based constitutive model is developed to predict the flow stress of the studied Al-Zn-Mg-Cu alloy during the hot compression tests. The correlation
12
coefficient (R) and the average absolute relative error (AARE) values between the
13
measured and predicted flow stresses are 0.993 and 4.855%, respectively. These
14
results reveal that the physically-based constitutive model has a good prediction
15
capability and can be used to determine the hot formation processing parameters for
16
Al-Zn-Mg-Cu alloys.
17
Acknowledgments
18
The authors express their appreciation for the financial support of National Natural
19
Science Foundation of China under Grant (No. 51405100), Shandong Provincial
20
Natural Science Foundation, China, (No. ZR2017PA003), Postdoctoral Science
21
Foundation of China under Grant (No. 2014M551233 and 2017T100237), Plan of
22
Key Research and Development in Shandong Province under Grant (No.
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ACCEPTED MANUSCRIPT 2017GGX202006), and Plan of Co-Development of University in Weihai under Grant
2
(No. 2016DXGJMS05).
3
Reference
4
[1] X. Huang, H. Zhang, Y. Han, W. Wu, J. Chen, Hot deformation behavior of 2026
5
aluminum alloy during compression at elevated temperature, Mater. Sci. Eng. A 527
6
(2010) 485-490.
7
[2] Y.C. Lin, L.T. Li, Y.C. Xia, Y.Q. Jiang, Hot deformation and processing map of a
8
typical Al-Zn-Mg-Cu alloy, J. Alloys Comp. 550 (2013) 438-445.
9
[3] H. Mirzadeh, M. Roostaei, M.H. Parsa, R. Mahmudi, Rate controlling mechanisms
M AN U
SC
RI PT
1
during hot deformation of Mg-3Gd-1Zn magnesium alloy: Dislocation glide and
11
climb, dynamic recrystallization, and mechanical twinning, Mater. Des. 68 (2015)
12
228-231.
13
[4] D.X. Wen, Y.C. Lin, Y. Zhou, A new dynamic recrystallization kinetics model for a
14
Nb containing Ni-Fe-Cr-base superalloy considering influences of initial δ phase,
15
Vacuum 141 (2017) 316-327.
16
[5] C.M. Sellars, W.J. McTegart, On the mechanism of hot deformation, Acta Mater.
17
14 (1966) 1136-1138.
18
[6] Y.C. Lin, S.C. Luo, L.X. Yin, J. Huang, Microstructural evolution and high
19
temperature flow behaviors of a homogenized Sr-modified Al-Si-Mg alloy, J. Alloys
20
Comp. 739 (2017) 590-599.
21
[7] L.X. Li, Y. Lou, L.B. Yang, D.S. Peng, K.P. Rao, Flow stress behavior and
22
deformation characteristics of Ti-3Al-5V-5Mo compressed at elevated temperatures,
AC C
EP
TE D
10
ACCEPTED MANUSCRIPT Mater. Des. 23 (2002) 451-457.
2
[8] A. Momeni, H. Arabi, A. Rezaei, H. Badri, S.M. Abbasi, Hot deformation behavior
3
of austenite in HSLA-100 microalloyed steel, Mater. Sci. Eng. A 528 (2011)
4
2158-2163.
5
[9] N. Haghdadi, A. Zarei-Hanzaki, H.R. Abedi, The flow behavior modeling of cast
6
A356 aluminum alloy at elevated temperatures considering the effect of strain, Mater.
7
Sci. Eng. A 535 (2012) 252-257.
8
[10] Y.C. Lin, M.S. Chen, J. Zhong, Prediction of 42CrMo steel flow stress at high
9
temperature and strain rate, Mech. Res. Commun. 35 (2008) 142-150.
M AN U
SC
RI PT
1
[11] T. Yan, E. Yu, Y. Zhao, Constitutive modeling for flow stress of 55SiMnMo
11
bainite steel at hot working conditions, Mater. Des. 50 (2013) 574-580.
12
[12] Y.C. Lin, X.M. Chen, D.X. Wen, M.S. Chen, A physically-based constitutive
13
model for a typical nickel-based superalloy, Comput. Mater. Sci. 83 (2014) 282-289.
14
[13] G. Ji, Q. Li, L. Li, A physically-based constitutive relation to predict flow stress
15
for Cu-0.4Mg alloy during hot working, Mater. Sci. Eng. A 615 (2014) 247-254.
16
[14] P.M. Souza, H. Beladi, R. Singh, B. Rolfe, P.D. Hodgson, Constitutive analysis
17
of hot deformation behavior of a Ti6Al4V alloy using physical based model, Mater.
18
Sci. Eng. A 648 (2015) 265-273.
19
[15] N. Hghdadi, D. Martin, P. Hodgson, Physically-based constitutive modelling of
20
hot deformation behavior in a LDX 2101 duplex stainless steel, Mater. Des. 106 (2016)
21
420-427.
22
[16] R. Zhu, Q. Liu, J. Li, S. Xiang, Y. Chen, X. Zhang, Dynamic restoration
AC C
EP
TE D
10
ACCEPTED MANUSCRIPT mechanism and physically based constitutive model of 2050 Al-Li alloy during hot
2
compression, J. Alloys Comp. 650 (2015) 75-85.
3
[17] J. Zuo, L. Hou, J. Shi, H. Cui, L. Zhuang, J. Zhang, Effect of deformation
4
induced precipitation on grain refinement and improvement of mechanical properties
5
AA 7055 aluminum alloy, Mater. Charact. 130 (2017) 123-134.
6
[18] G. Chen, F.Y. Lin, S.J. Yao, F. Han, B. Wei, Y.M. Zhang,
7
of aluminum alloy in a wide temperature range from warm to semi-solid regions, J.
8
Alloys Comp. 674 (2016) 26-36.
9
[19] B. Wu, M.Q. Li, D.W. Ma, The flow behavior and constitutive equations in
10
isothermal compression of 7050 aluminum alloy, Mater. Sci. Eng. A 542 (2012)
11
79-87.
12
[20] G. Meng, B.L. Li, H.M. Li, H. Huang, Z.R. Nie, Hot deformation and processing
13
maps of an Al-5.7 wt.% Mg alloy with erbium, Mater. Sci. Eng. A 517 (2009)
14
132-137.
15
[21] X.Y. Liu, Q.L. Pan, Y.B. He, W.B. Li, W.J. Liang, Z.M. Yin, Flow behavior and
16
microstructural evolution of Al-Cu-Mg-Ag alloy during hot compression deformation,
17
Mater. Sci. Eng. A 500 (2009) 150-154.
18
[22] S. Gourdet, F. Montheillet, A model of continuous dynamic recrystallization,
19
Acta Mater. 51 (2003) 2685-2699.
20
[23] D.D. Chen, Y.C. Lin, Y. Zhou, M.S. Chen, D.X. Wen, Dislocation substructures
21
evolution and an adaptive-network-based fuzzy inference system model for
22
constitutive behavior of a Ni-based superalloy during hot deformation, J. Alloys
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Constitutive behavior
ACCEPTED MANUSCRIPT Comp. 708 (2017) 938-946.
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[24] N. Haghdadi, P. Cizek, H. Beladi, P.D. Hodgson. A novel high-strain-rate ferrite
3
dynamic softening mechanism facilitated by the interphase in the austenite/ferrite
4
microstructure, Acta Mater. 126 (2017) 44-57.
5
[25] M.R. Rokni, A. Zarei-Hanzaki, A.A. Roostaei, A. Abolhasani, Constitutive base
6
analysis of a 7075 aluminum alloy during hot compression testing, Mater. Des. 32
7
(2011) 4955-4960.
8
[26] N. Jin, H. Zhang, Y. Han, W. Wu, J. Chen, Hot deformation behavior of 7150
9
aluminum alloy during compression at elevated temperature, Mater. Charact. 60 (2009)
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530-536.
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[27] E. Cerri, E. Evangelista, A. Forcellese, H.J. McQueen, Comparative hot
12
workability of 7012 and 7075 alloys after different pretreatments, Mater. Sci. Eng. A
13
197 (1995) 181-198.
14
[28] E.P. Busso, F.A. Mcclintock, A dislocation mechanics-based crystallographic
15
model of a B2-type intermetallic alloy, Int. J. Plast. 12 (1996) 1-28.
16
[29] C. Zener, J.H.Hollomon, Effect of strain rate upon plastic flow of steel, J. Appl.
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Phys. 15 (1944) 22-32.
18
[30] J. Jonas, C.M. Sellars, W.J. Tegart, Strength and structure under hot-working
19
conditions, Int. Mater. Rev. 14 (1969) 1-24.
20
[31] H. Shi, A.J. Mclaren, C.M. Sellars, R. Shahani, R. Bolingbroke, Constitutive
21
equations for high temperature flow stress of aluminium alloys, Mater. Sci. Technol.
22
13 (1997) 210-216.
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ACCEPTED MANUSCRIPT [32] M. Zhou, Y.C. Lin, J. Deng, Y.Q. Jiang, Hot tensile deformation behaviors and
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constitutive model of an Al-Zn-Mg-Cu alloy, Mater. Des. 59 (2014) 141-150.
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[33] A. Laasraoui, J.J. Jonas, Prediction of steel flow stresses at high temperatures and
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strain rates, Metall. Mater. Trans. A 22 (1991) 1545-1598.
5
[34] X.M. Chen, Y.C. Lin, D.X. Wen, J.L. Zhang, M. He, Dynamic recrystallization
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behavior of a typical nickel-based superalloy during hot deformation, Mater. Des. 57
7
(2014) 568-577.
8
[35] N. Haghdadi, A. Zarei-Hanzaki, A.R. Khalesian, H.R. Abedi. Artificial neural
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network modeling to predict the hot deformation behavior of an A356 aluminum alloy,
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Mater. Des. 49 (2013) 386-391.
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Figure Captions
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Fig. 1 Initial microstructure of the extruded Al-Mg-Zn-Cu alloy sample.
Fig. 2 Microstructures of the studied Al-Mg-Zn-Cu alloy deformed at (a) 350 °C, (b)
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400 °C, (c) 450 °C, and (d) 470 °C with a strain rate of 0.01 s-1.
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Fig. 3 Microstructures of the studied Al-Mg-Zn-Cu alloy deformed at 470 °C with
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strain rates of (a) 1 s-1, (b) 0.1 s-1, (c) 0.01 s-1, and (d) 0.001 s-1.
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Fig. 4 TEM images of the studied Al-Mg-Zn-Cu alloy deformed at (a) 350 °C/0.001
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s-1, (b) 470 °C/0.1 s-1, and (c) 470 °C /0.001 s-1.
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Fig. 5 True strain-stress curves at different strain rates: (a) 0.001 s-1, (b) 0.01 s-1, (c)
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0.1 s-1, and (d) 1 s-1.
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Fig. 6 A schematic diagram of true stress-strain curves (Symbols ‘a’ and ‘b’ show the
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dynamic recovery and dynamic recrystallization mechanisms, respectively.).
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Fig. 7 (a) Schematic of the θ-σ curves for the studied Al-Mg-Zn-Cu alloy at various
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temperatures and strain rates. (b) The θ-σ curve at a temperature 470 °C with a strain
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rate of 0.001 s-1.
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Fig. 9 Linear relationship fitting of ln(sinh(ασsat))- ln ε& .
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Fig. 8 The relationships between (a) lnσsat and ln ε& , and (b) σsat and ln ε& .
Fig. 10 Linear relationship fitting of ln(sinh(ασsat))-1/T.
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Fig. 11 Relationship between the yield stress (σ0) and the Zener-Hollomon parameter.
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Fig. 12 Relationship between the coefficient of dynamic recovery (Ω) and the
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Zener-Hollomon parameter.
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Fig. 13 Relationship between the steady stress (σss) and the Zener-Hollomon
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parameter.
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Fig. 14 Relationship between the peak strain (εp) and the Zener-Hollomon parameter.
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Fig. 15 Relationship between the materials constants (kd) and the Zener-Hollomon
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parameter.
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Fig. 16 Comparisons between the predicted and measured flow stresses of the studied
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Al-Mg-Zn-Cu alloy at temperatures of (a) 350 °C, (b) 400 °C, (c) 450 °C, (d) 470 °C,
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and (e) 490 °C.
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Fig. 17 Correlation between the predicted and experimental flow stress data.
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Fig. 1 Initial microstructure of the extruded Al-Mg-Zn-Cu alloy.
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Fig. 2 Microstructures of the studied Al-Mg-Zn-Cu alloy deformed at (a) 350 °C, (b)
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400 °C, (c) 450 °C, and (d) 470 °C with a strain rate of 0.01 s-1.
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Fig. 3 Microstructures of the studied Al-Mg-Zn-Cu alloy deformed at 470 °C with
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strain rates of (a) 1 s-1, (b) 0.1 s-1, (c) 0.01 s-1, and (d) 0.001 s-1.
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Fig. 4 TEM images of the studied Al-Mg-Zn-Cu alloy deformed at (a) 350 °C/0.001
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s-1, (b) 470 °C/0.1 s-1, and (c) 470 °C /0.001 s-1.
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Fig. 5 True strain-stress curves at different strain rates: (a) 0.001 s-1, (b) 0.01 s-1, (c)
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Fig. 6 A schematic diagram of true stress-strain curves (Symbols ‘a’ and ‘b’ show the
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dynamic recovery and dynamic recrystallization mechanisms, respectively.).
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Fig. 7 (a) Schematic of the θ-σ curves for the studied Al-Mg-Zn-Cu alloy at various temperatures and strain rates. (b) The θ-σ curve at a temperature 470 °C with a strain
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Fig. 8 The relationships between (a) lnσsat and ln ε& , and (b) σsat and ln ε& .
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Fig. 9 Linear relationship fitting of ln(sinh(ασsat))- ln ε& .
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Fig. 10 Linear relationship fitting of ln[sinh(ασsat)]-1/T.
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Fig. 11 Relationship between the yield stress (σ0) and the Zener-Hollomon parameter.
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Fig. 12 Relationship between the coefficient of dynamic recovery (Ω) and the
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Zener-Hollomon parameter.
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Fig. 13 Relationship between the steady stress (σss) and the Zener-Hollomon
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Fig. 14 Relationship between the peak strain (εp) and the Zener-Hollomon parameter.
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Fig. 15 Relationship between the materials constants (kd) and the Zener-Hollomon
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Fig. 16 Comparisons between the predicted and measured flow stresses of the studied Al-Mg-Zn-Cu alloy at temperatures of (a) 350 °C, (b) 400 °C, (c) 450 °C, (d) 470 °C, and (e) 490 °C.
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Fig. 17 Correlation between the predicted and experimental flow stress data.
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The hot deformation behavior of an Al-Zn-Mg-Cu alloy was studied by isothermal hot compression tests.
The hot deformation activation energy of the Al-Zn-Mg-Cu alloy was estimated to be 186 kJ/mol.
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The physically-based constitutive model of the Al-Zn-Mg-Cu alloy was
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