Multi-mode distortion behavior of aluminum alloy thin sheets in immersion quenching

Journal Pre-proof Multi-mode distortion behavior of aluminum alloy thin sheets in immersion quenching Z.X. Li (Conceptualization) (Methodology) (Investigation) (Writing original draft), M. Zhan (Writing - review and editing) (Supervision) (Project administration) (Funding acquisition), X.G. Fan (Conceptualization) (Writing - review and editing) (Funding acquisition), X.X. Wang (Investigation), F. Ma (Investigation), R. Li (Investigation)

PII:

S0924-0136(19)30549-7

DOI:

https://doi.org/10.1016/j.jmatprotec.2019.116576

Reference:

PROTEC 116576

To appear in:

Journal of Materials Processing Tech.

Received Date:

3 July 2019

Revised Date:

7 December 2019

Accepted Date:

25 December 2019

Please cite this article as: Li ZX, Zhan M, Fan XG, Wang XX, Ma F, Li R, Multi-mode distortion behavior of aluminum alloy thin sheets in immersion quenching, Journal of Materials Processing Tech. (2019), doi: https://doi.org/10.1016/j.jmatprotec.2019.116576

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Multi-mode distortion behavior of aluminum alloy thin sheets in immersion quenching

Z.X. Lia, M. Zhana *, X.G. Fana, X.X. Wanga, F. Mab, R. Lia

a

State Key Laboratory of Solidification Processing, Shaanxi Key Laboratory of High-Performance

Northwestern Polytechnical University, Xi’an 710072, China b

Long March Machinery Factory, China Aerospace Science and Technology Corporation, Chengdu,

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610100, China

Corresponding Author. Tel.: +86-029-88460212-801; Fax: +86-029-88495632; Email:

Abstract

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[email protected] (M. Zhan)

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*

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Precision Forming Technology and Equipment, School of Materials Science and Engineering,

Aluminum alloy thin sheets are widely used to produce light-weight

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high-strength components and the sheets are generally quenched before forming (Q-F) to improve the final mechanical properties of components. However, the distortion in

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quenching will significantly affect the quality and stability of the following forming process. In this study, the distortion behavior of 2219 aluminum alloy thin sheets and its forming mechanism in quenching process were investigated through experiments and finite element analysis (FEA). The results indicate that, the quenched thin sheets basically present three distortion modes, namely saddle-shape, shovel-shape and arch-shape. The distortion modes of the quenched thin sheets are determined by the bending modes of distortion zone which is the zone of the thin sheet near the water 1 / 61

surface. When the bending mode is one-half sine wave, the quenching thin sheets always show saddle-shape. However, for the cases where the bending mode changes to three-half sine wave and the bending of distortion zone is suppressed, the quenching thin sheets show shovel-shaped and arch-shaped distortion modes, respectively. The reasons behind the variations of the bending mode are finally analyzed based on a buckling criterion under laterally constrained conditions. These results will provide an in-depth understanding of quenching distortion behavior and lay a basic guidance for controlling the distortion of thin sheets in quenching process.

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Keywords: Quenching distortion behaviors, Aluminum alloy thin sheets, Immersion quenching, Finite element analysis. 1. Introduction

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Aluminum alloy thin-walled components are widely used in the aerospace and automobile industries due to their lightweight, low energy consumption and high

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stability (Dursun and Soutis, 2014). Heat-treatable strengthening aluminum alloy thin sheets are important raw materials for forming such components. According to the

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order of hot and cold working, there are two processing routes to manufacture large thin-walled components. The first route is forming-quenching (F-Q), namely, the required shape is obtained by metal forming with subsequent quenching (Li et al.,

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2013). However, in the quenching process, the thin-walled components are prone to occurring severe distortion, thus affecting its final shape and dimensional accuracy

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(Nallathambi et al., 2008). Another route is quenching-forming (Q-F) which means the sheet can be quenched before the forming process (Khan et al., 2018). This route

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can obtain a component with good geometric accuracy and mechanical properties (Wang and Huang, 2016). However, the distortion produced in the quenching process makes the state of the thin sheets deviate from the ideal situation, i.e., shape distortion, initial residual stress, and these deviations significantly affect the stability of the following forming process. Therefore, in order to optimize the forming process it is necessary to know the state of the thin sheet after quenching (Weiss et al., 2012). Finite element analysis (FEA) is a good method to study the quenching process. 2 / 61

Using the FEA, some researchers investigated the immersion quenching process of medium and thick aluminum sheets. Their FE simulations indicated that the thick sheets became drum-shaped after quenching under the action of thermal stress (Li et al., 2016), and the residual stress distributed symmetrically along the middle surface of the thick sheets (Koç et al., 2006). These results were in good agreement with experimental results under the two assumptions: (1) The thick sheets do not suffer distortion instability during quenching; (2) All of the surfaces of the thick sheets are immersed into the quenchant at the same time, that is, the cooling process during

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immersing process is ignored. However, for the thin sheets with low rigidity, the distortion instability phenomenon is very prone to occur (Totten and Mackenzie,

2000), and thus the quenching residual stress distribution becomes more complicated in the thin sheets than that in the thick sheets. In addition, because of fast cooling

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speed of thin sheets, the immersing process could result in a large temperature

difference between areas immersed into quenchant at different moments. Therefore,

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the cooling process during immersing process significantly affects the temperature distribution of thin sheet and should be considered in study of the quenching

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distortion behavior of the thin sheets.

Up to now, some researchers investigated the mechanisms of quenching

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distortion. Silva et al. (2012) investigated the quenching of an AISI 4140 steel C-ring, and their FE simulations and experiments showed that the geometric distortion of the C-ring was associated with the phase transformation at the thickest part of the ring

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during the final stages of the quenching process. Narazaki et al. (2012) found that the bending and warping of long thin steel parts with uneven thickness were mainly

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resulted from longitudinal strain that largely depends on non-uniformity of cooling parts in quenching. However, these investigations only ascribed the quenching distortion to phase transformation in quenching or shape asymmetry of components. For aluminum alloy thin sheets, even if the geometrical shape is symmetrical and no phase transformation occurs during quenching, distortion is also observed (Totten and Mackenzie, 2000). This distortion is considered as the result of structural instability of thin sheets under the action of thermal stress caused by quenching, and the distortion 3 / 61

instability behavior need to be studied. Using implicit FEA, the quenching thermal stress can be predicted accurately (Koç et al., 2006). However, as quenching distortion instability occurs, the element stiffness matrix is singular at the bifurcation point and the solution cannot be further carried out. Therefore, In order to capture instability behavior during quenching, the discontinuous bifurcation problem usually can be changed to a non-linear continuous response problem by introducing initial geometric imperfections into the initial geometry model (Liu et al., 2014). There are two main methods to introduce initial

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geometric imperfections. For the first methods, the initial imperfections are introduced by seeding a specific deflection function into the geometric model

(Paquette and Kyriakides, 2006), which usually are used to obtain the instability

behavior under simple loading conditions. However, the distortion in the quenching

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process is resulted from complex internal thermal stress and internal strain caused by the temperature variation and the phase transformation. The thermal stress and strain

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distribution characteristics also change during over time. This change can affect the quenching distortion process, which could result in the final distortion mode after

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quenching is different from the initial instability mode. Therefore, it is difficult and unreasonable to determine a possible initial deflection function before quenching, and

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it could achieve a more realistic result to use another method of seeding random initial defections (Papadopoulos and Papadrakakis, 2005). Cao and Boyce (1997a) introduced the random initial imperfections to initiate wrinkling in the study of deep

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drawing operations by using randomly distributed rows of thinned and offset elements. Gusic et al. (2000) used the static nonlinear buckling analysis with considering the

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initial geometrical microdefects to investigate the buckling behavior of the cylindrical shell structure. Through the analysis, the sensitivity of the shell structure to geometric defects was obtained. Papadopoulos and Papadrakakis (2005) introduced the random imperfect geometry of shell structures as well as the modulus of elasticity into a perfect geometry model to investigate the effect of material and thickness imperfections on the buckling load of isotropic shells. For the quenching distortion process, because of the complicated thermal 4 / 61

boundary conditions and the varying material property with temperature, the instability behavior is more complicated. However, up to now, there are little literatures that investigate the quenching distortion behaviors of aluminum alloy thin sheets. To this end, a finite element model of quenching considering immersion cooling process of thin sheets with random initial imperfections was established in this study. Based on the FE model, the quenching distortion mechanism of thin sheets was revealed. It is expected that the results of these investigations would be useful for

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designing appropriate Q-F procedure, and for controlling the quenching distortion of thin sheets. 2 Experimental details

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The material used in this study was a cold rolled 2219 aluminum alloy thin sheet with the thickness (t) of 1.5 mm, 2 mm and 3 mm, respectively. The chemical

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compositions of the alloy (wt%) were listed in Table 1. The thin sheet was 200× 200mm square with a T-shaped clamping end, as shown in Fig. 1 (a). The T-shaped

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clamping end was used to fix the sheet and weaken the influence of the constraint on the quenching distortion. The ratio of thickness and length (thickness ratio t/a) for the

respectively.

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1.5 mm, 2mm and 3 mm thickness thin sheet was 0.0075, 0.01 and 0.015,

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Table 1 Main chemical compositions of 2219 aluminum alloy. Element Cu Mn Fe Zr Ti Si Zn 6.5

0.36

0.21

0.18

0.06

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wt.%

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0.05

0.02

Mg

Al

0.01

Bal.

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Fig. 1. Quenching experiments: (a) initial shape of thin sheet and (b) schematic illustration of quenching process.

A water quenching apparatus which allows controlled immersing speed and in-situ observation of distortion was designed, as shown in the Fig. 1 (b). In

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quenching experiment, the thin sheets were first heated to 530 ℃ for 2 hours in order to achieve full solution treatment. Then the sheets were quickly transferred to the

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apparatus and placed on a pneumatic lifting system which was used to immerse the sample into water at a specific speed. The time of transfer process from the heating

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furnace to quenching tank was about 5 s. The initial temperature of water was 25±1◦C. The immersing speed can be adjusted by rotating the throttle valve. To observe the evolution of thin sheets during quenching, the quenching process was recorded at 0.2s

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intervals by a high-resolution camera. 3 Numerical simulation

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To investigate the distortion behaviors of thin sheets in the quenching process, the FE simulation of the process was conducted. This process requires dealing with

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both a transient heat conduction problem and a thermal viscoelasticity problem. In order to capture the instability behavior of thin sheets during quenching, the initial thickness imperfections were considered in the FE modeling. 3.1 Mathematical model During quenching, the transient temperature field of workpieces can be calculated by using the Fourier heat conduction equation as follows: 6 / 61

c

T  div  k  gradT   Q t

(1)

where  , c and T are the density, specific heat, and transient temperature, respectively. Q is the latent heat of phase transformation. For the aluminum alloy, the aluminum matrix (face-centered-cubic, FCC) does not undergo phase transformation, and the formation of precipitates can be inhibited effectively in quenching. Thus, the latent heat of phase transformation is neglected, and Q is considered as 0. For the temperature distribution of initial condition, i.e., T0, it can be written as:

T t 0  T0  x, y, z 

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(2)

Boundary condition belongs to the third type condition, which represents the heat exchange (radiation and convection) of quenching workpiece with ambience. It can be

λ

T  s  h Tw  Tc n

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described as:



(3)

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where  is thermal conductivity; h is the total heat transfer coefficient (HTC), and

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it can be obtained by the inverse heat conduction analysis based on the FEA; Tc is the quenchant temperature that is usually known as a constant; Tw is the surface temperature of workpiece.

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Quenching distortion can be considered as a thermal viscoelasticity problem with temperature loading. The constitutive equations for simulation of quenching is usually

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defined in the form of the additive decomposition of the strain rate tensors, and the total strain rate is the sum of elastic, plastic and thermal strain rate for aluminum alloy,

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showing in Eq. (4).

 ij   ije   ijp   ijth

(4)

where  ij ,  ije ,  ijp and  ijth are the total, elastic, plastic and thermal strain rate. 3.2 Inverse determination of boundary conditions In order to obtain the boundary conditions of actual heat exchange between water and aluminum alloy thin sheets during quenching, the total HTC should be 7 / 61

determined. In this section, the HTC of the 2219 aluminum alloy thin sheet was inversely calculated by measuring the cooling curve with the help of FE package Deform-HT (Sugianto et al., 2008). To simulate the actual cooling process, a specific temperature probe was designed according to the material and shape of the actual workpieces (Maniruzzaman and Sisson, 2004) and the initial temperature of the probe is close to the actual quenching start temperature (Maniruzzaman et al., 2012). The illustration of the designed sheet probe is shown in Fig. 2 (a). A Φ0.6 mm× 40mm cylindrical hole was drilled at the center of sheet to place a K-type

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thermocouple with a 0.5 mm diameter. Before inserting the thermocouple, a sealant that has the similar physical properties with aluminum alloy was applied on the

surface of thermocouple to prevent water from penetrating. During quenching, the probe was cooled from initial temperature of 540 ℃ in water at 25 ℃, and the

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temperatures of the “T” point were recorded using the temperature collecting system. The measuring frequency of the sensor temperature was 50 Hz.

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Based on the measured water cooling curve, a temperature dependent water-sheet HTC curves of 2219 aluminum alloy sheet were calculated, as shown in Fig. 2 (b).

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The air-sheet HTC between air and sheet at the temperatures of 530-400℃ was also measured to analyze the cooling of sheet in the transfer process. The results indicated

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that the air-sheet HTC can be regarded as a constant of 0.02 KW/m2℃.

Fig. 2. Measurement of HTC: (a) illustration of sheet probe, (b) measured water cooling curve and inversely determined HTC curve. 8 / 61

3.3 Characterization of initial thickness imperfections The thin sheets always have inevitable imperfection during the forming process. In the present work, the thickness of the thin sheet was measured using an in-hand ultrasonic thickness-measuring instrument (PX-7DL). The thin sheet was equally divided into 25 small square pieces firstly. Then the wall thickness at the center of each small piece was measured. The thickness distribution was obtained by linear interpolation of the data.The measured actual thickness distribution was shown in Fig.

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3, and it can be determined that the thickness deviation of the experimental thin sheet

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was ±0.01 mm.

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Fig. 3. Thickness distribution of the initial sheet.

3.4 Finite element modeling for quenching process An FE model for the quenching process was established using

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ABAQUS/Standard (version 6.14) platform. In the model, all of input material properties were taken from the experiments. The true stress-strain curves at different

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temperatures and strain rates were obtained based on the standard tensile tests in our previous work (Li et al., 2018). The yield stress was determined using the 0.2% offset approach. The thermal parameters and yield strength of the as-quenched 2219 aluminum alloy used in this study are listed in Tables 2 and 3, respectively. In the quenching process, the thin sheets with a uniform temperature of 530℃ were immersed in water of 25℃ at different immersing speeds after 5 second air cooling 9 / 61

until it was cooled down to a temperature below 30℃. The thin sheet was meshed by an 8-node linear heat transfer brick element (DC3D8) and an 8-noded linear reduced integration brick element (C3D8R) in the simulations for heat transfer analysis and stress/distortion analysis, respectively (Fig. 4). The total number of element was

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40000 with 100 along the side and 4 through the thickness.

Table 2 Main thermal parameters of 2219 aluminum alloy at different temperatures.

22.707 24.246 25.989 26.587 26.604 26.116

103.793 109.553 122.849 136.363 138.533 122.685

Specific heat (J/g/K)

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T (℃)

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Thermal conductivity (W/m·k)

1.137 1.151 1.197 1.263 1.297 1.345

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Thermal expansion (10-6K-1)

25 100 200 300 400 530

Table 3 Elastic modulus and yield strength at different temperatures.

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75.890

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72.752

65.891

58.518

44.973

Yield strength (MPa)

Strain rate (s-1)

172 165 158 160 152 151 128 136 141 109 128 141 58 73

0.001 0.01 0.1 0.001 0.01 0.1 0.001 0.01 0.1 0.001 0.01 0.1 0.001 0.01

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Elastic modulus (GPa)

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T (℃)

25

100

200

300

400

28.652

0.1 0.001 0.01 0.1

530

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88 7 11 19

Fig. 4. Meshed geometry used for the simulation.

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To consider the random initial imperfection of sheet, the experimental thickness

fluctuations were seeded in the geometric model by deviating the coordinate of sheet surface node along the thickness direction. Therefore, the deviated coordinate

f  x, y  2

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Z  x, y   Z 0 

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Z  x, y  can be calculated by the following function:

(5)

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where Z 0 is the coordinate value of the ideal thin sheet surface in the thickness direction, and f  x, y  is the random field modeling the thickness fluctuation of

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±0.01 mm.

In order to simulate the immersing process, the thin sheet was divided into

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several small heat transfer zones whose width was equal to the width of one element, showing in Fig. 4. Through this division, different zones of the sheet surfaces can be set as different heat transfer conditions. If a zone is immersed into water, the HTC of this zone is set as water-sheet HTC, otherwise set as air-sheet HTC. After immersing process, the thin sheets were still soaked in water until its temperature was close to the water temperature of 25 ℃. This soak time was set as 20 seconds. According to the number of small heat transfer zones, several analysis steps were 11 / 61

set, and the time of each analysis step was calculated by Eq. (6). For every analysis step, the automatic time step control techniques (ABAQUS, 2014) for the solution of transient problems were used.

t 

L n

(6)

where L is the width of thin sheets; n is the number of small heat transfer zone; v is the average immersing speed. Using the FE model established above, the simulation of quenching process was

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performed on a computer machine equipped with an Intel(R) Xeon(R) Gold 6130 CPU @ 2.10GHz and the Windows(R) 10 (64-bit platform) operating system. The

detailed simulation procedure is summarized, as the flowchart shown in Fig. 5. First,

the time-dependent temperature fields were calculated for different immersing speeds.

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Then using the temperature fields as the input condition, the full responses of the structure were computed by the arc length global solution method. However, in

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quenching process, a localised snap-through in quenching thin sheet would occur under the action of changing thermal stress, and the global solution methods may not

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work. Thus, in order to solve this problem, an artificial viscous force, Fv , (Eq. (7)) and the corresponding numerical damping is introduced into the global equilibrium

Fv  cM *v

(7) (8)

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P  I  Fv  0

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equation (Eq. (8)) at all nodes in nonlinear quasi-static procedure (ABAQUS, 2014).

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where M * is the artificial mass matrix calculated with unity density, c is the damping factor, v is the vector of nodal velocities, I is the internal force, P is the external force.

The damping factor, i.e., the stabilize factor, is controlled by the convergence

history and the ratio of the energy dissipated by viscous damping to the total internal energy, and has a default value of 2×10−4. It can be known from Eq. (8) that as the damping factor increases, the ratio of the energy dissipated by viscous damping to the 12 / 61

total internal energy increases, and it can increase the deviation between the computed equilibrium path and the actual path. To ensure the simulation accuracy, the ratio is limited by an accuracy tolerance that is required to be less than 5% (ABAQUS, 2014). If the ratio exceeds the accuracy tolerance, the damping factor is need to be reduced to ensure that the ratio is less than the accuracy tolerance. Using the above automatic stabilization scheme, the damping factor can be made very small without having a major effect on the solution (Wong and Pellegrino, 2006). Based on this guideline, the

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possible lower damping factor determined in the present study was 2×10-5.

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Fig. 5. Flowchart for quenching analysis.

For the purpose of accurately monitoring the distortion degree, the relative

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coordinates of every quenched thin sheet edge were measured using a three coordinate measuring machine (GLOBAL STATUS121510). Using a line connecting the two endpoints of one edge as the datum line, the bending height (BH) can be calculated through the distance from the edge to the baseline, as shown in Fig. 6 (b) and (c). Meanwhile, as shown in Fig. 6 (a), the bottom, upper and side edges represents the edges that were first submerged into the water, finally submerged into the water, and perpendicular to the water, respectively. Since the bending directions of these edges 13 / 61

may be different, the BH was defined as positive value when the bending direction was the same as that of the bottom edge (Fig. 6 (b)), otherwise was defined as negative value (Fig. 6 (c)). Because the two side edges had the same distortion for the same sheet, only one side edge was chosen to measure. For the simulation results, the BHs of the three edges can also be calculated according to the node coordinates (Fig.

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6 (a)).

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Fig. 6. Measurement of BH: (a) the bottom, upper and side edges of thin sheet, and (b) positive BH and (c) negative BH for bottom edge and side edge, respectively.

4. Quenching distortion analysis and FE model verification In this section, the distortion modes of quenched thin sheets with different

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immersing speeds and different thickness ratios were studied. Based on the experimental results, the typical distortion modes were analyzed. To verify predictive

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capability of this FE model, the quenching distortion degree of every typical distortion mode obtained from experiments and FE simulations was compared.

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4.1 Effect of immersing speed on quenching distortion The thin sheets with 0.01 thickness ratio were chosen to investigate the effect of

immersing speed on quenching distortion. Fig. 7 shows the experimental results of quenching distortion with different immersing speeds. At the immersing speed of 0.50 m/s, 0.32 m/s and 0.16 m/s, the bottom and upper edges are bent in width along the same direction (concave upward bending in Fig. 7), and the side edges are bent in length along the opposite direction (concave downward bending in Fig. 7). Thus, the 14 / 61

quenched thin sheets show the saddle-shaped distortions at this speed region. Meanwhile, it also can be observed that the bending degree of the all edges increases with the decrease of immersing speed. At the immersing speed of 0.10 m/s, the bending degree of the bottom edge becomes very small while the bending degree of the upper edge continues to increasing. Furthermore, the bending direction of upper edge (concave downward bending in Fig. 7) is opposite to that of bottom edge. This type of distortion can be classified as the second mode named the shovel-shaped distortion.

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When the immersing speed decreases to 0.05 m/s, the third distortion mode of arch-shaped distortion (showing in Fig. 7) is observed. In this case, the bottom and

upper edges bend slightly while the shape of the side edges show obvious arch-shaped. With the immersing speed decreasing to 0.03 m/s, the distortion mode no longer

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changes and the distortion degree tends to be stable.

In summary, the quenched thin sheets exhibit three distortion modes as follows:

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(1) saddle-shaped mode: the bending direction of the upper edge is the same as that of bottom edge, that is, the bending height of upper edge is positive; (2) shovel-shaped

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mode: the upper and bottom edges are bent to the opposite direction, that is, the bending height of the upper edge is negative; (3) arch-shaped mode: the upper edge or

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bottom edge is almost straight, that is, the bending height of the upper edge or bottom

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edge is closed to zero.

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4.2 Effect of thickness ratio on quenching distortion

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Fig. 7. Images of thin sheets after quenching.

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The quenched thin sheets with different thickness ratios are compared in Fig. 8. It can be seen that the total distortion degree of the thin sheets decreases with the

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increase of thickness ratio at the same immersing speed. In the case that the thickness ratio is 0.0075, the bending height of upper edges and bottom edges are either negative or positive (shown from the bending height in Fig. 8), and the zero bending

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height is not observed, thus only two distortion modes (shown from the images of thin sheets in Fig. 8), i.e., saddle-shaped and shovel-shaped modes, can be observed.

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When the thickness ratio of the thin sheets increases to 0.015, the bending heights of upper edge and bottom edge are positive at all speed, which indicates that the thin

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sheets only exhibit the saddle-shaped mode.

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Fig. 8. Distortion modes of quenched thin sheets with different thicknesses ratios (t/a).

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4.3 Comparison of experiment and simulation for typical distortion modes

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From the experimental results in Section 4.1 and 4.2, it can be known that the quenched thin sheets basically exhibited three distortion modes, i.e., saddle-shape, shovel-shape and arch-shape, at different immersing speeds and different thickness

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ratios. Because all distortion modes were observed for the quenched thin sheet with 0.01 thickness ratio, this condition was chosen to verify the reliability of the proposed

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FE model. As shown in Fig. 9, three quenching distortion modes of thin sheets were also observed in the FE analyses at different speeds. The BH obtained from

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experiment and simulation is close and the average absolute error in the worst case is about 30%. This deviation may be caused by experimental errors, e.g., the immersing direction is not perpendicular to the water surface and the surface condition of sheet is non-uniform, and model simplifications, such as the ignorance of the water flow and other factors that could result in the heat fluctuations.

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18

(a)

6 0

-12 0

200

24 (b) 18 FEM simulation-0.32m/s Exp.-0.32m/s FEM simulation-0.10m/s Exp.-0.10m/s 12 FEM simulation-0.05m/s Exp.-0.05m/s 6 0 -6 -12 -18 -24 0 40 80 120 160 200 Upper edges-distance (mm) 18

(c)

12

BH (mm)

40 80 120 160 Botton edges-distance/mm

Exp.-0.32m/s Exp.-0.10m/s Exp.-0.05m/s

6

FEM simulation-0.32m/s FEM simulation-0.10m/s FEM simulation-0.05m/s

0 -6 -12 0

Immersing direction 40 80 120 160 Side edges-distance (mm)

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-18

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BH (mm)

-18

FEM simulation-0.32m/s FEM simulation-0.10m/s FEM simulation-0.05m/s

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Exp.-0.32m/s Exp.-0.10m/s Exp.-0.05m/s

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-6

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BH (mm)

12

200

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Fig. 9. Comparison of BHs between predicted results and experimental results: (a) bottom edges, (b) upper edges and (c) side edges.

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5. Multi-mode quenching distortion behaviors In this section, the thin sheet with 0.01 thickness ratio was chosen to study the

quenching distortion behavior. Through in-situ observation, the quenching distortion evolution of three modes was analyzed and the underlying distortion mechanisms were further explored. 5.1 Evolution of distortion To investigate the distortion behavior during quenching, the details of distortion 18 / 61

evolution at different immersing speeds were analyzed by in-situ observation. In these cases, both of the shapes obtained from simulation and experiment at the same moment of quenching process were given to make a comparison. In the simulation results, the colours indicate the contour maps of Mises stress distribution. It is observed that the shapes obtained from simulation are well agreement with experimental results, as shown in Figs. 10-12, which indicates that the simulation can successfully track the quenching distortion processes. At a fast immersing speed such as 0.32 m/s (Fig. 10), the thin sheet still keeps

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plane shape when 30% sheet is immersed into water (Fig. 10 (a)). Then slight bending occurs in width when about 50% is immersed into water (Fig. 10 (b)). After the

immersion depth increases to 80%, an obvious saddle-shaped distortion was observed. As the distortion degree further increases, the bottom edge and the upper edge are

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gradually bent along the same direction (Fig. 10 (c)), and this distortion shape keeps

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when the whole sheet is immersed into water (Fig. 10 (d)).

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Fig. 10. Shape variation of thin sheet under immersing speed of 0.32 m/s with immersion depth: (a) 30%, (b) 50%, (c) 80% and (d) 100%.

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At the immersing speed of 0.10 m/s (Fig. 11), the bottom edge of the thin sheet is obviously bent while its upper edge is bent slightly to the opposite direction (Fig. 11 (a)) when 30% of thin sheet is immersed into water. With the increase of immersion depth, the bending degree of bottom edge and upper edge continues to increase (Fig. 11 (b)) until the occurrence of snapping in the upper part of thin sheet. After that, the bending direction of the side edge suddenly changes from convex rightward to convex leftward showing in Fig. 11 (c). Moreover, the bending degree of bottom edge 19 / 61

obviously decreases while that of the upper edge increases with the increase of immersing depth, and the bending directions of the upper edge and bottom edge are opposite (Fig. 11 (c) and (d)). Finally, the quenched thin sheet exhibits shovel-shaped

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distortion (Fig. 11 (d)).

Fig. 11. Shape variation of thin sheet under immersing speed of 0.10 m/s with immersion depth: (a)

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30%, (b) 50%, (c) 60% and (d) 100%.

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When the immersing speed decreases to 0.05 m/s (Fig. 12), at the early quenching stage (about 40% immersion depth), the bottom edge and upper edge are

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bent obviously along the same direction and the thin sheet exhibits a saddle-shaped distortion (Fig. 12 (a)). As the immersion depth increases, the bending degree of the bottom edge and upper edge gradually decreases, and the edges becomes straight

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finally (Fig. 12 (b) and (c)). Therefore, after quenching, only a notable side bend is

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observed and the quenched thin sheet shows an arch-shape (Fig. 12 (d)).

Fig. 12. Shape variation of thin sheet under immersing speed of 0.05 m/s with immersion depth: (a) 40%, (b) 55%, (c) 80% and (d) 100%. 20 / 61

5.2 Mechanism of distortion Based on the analyses in Section 5.1, it is known that the multiple distortion modes of thin sheets result from different evolution processes of quenching distortion. In this section, the mechanism of the distortion onset and growth for the thin sheet in quenching was discussed with the help of the simulation. 5.2.1 Onset of distortion

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Generally, the inhomogeneous temperature distribution is the intrinsic reason for quenching distortion (Narazaki et al., 2012). In order to understand the onset and

growth of quenching distortion, temperature variation and its effect on distortion of thin sheet should be analyzed first.

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Taking the temperature distribution at 0.10 m/s immersing speed and 50%

immersion depth as an example (Fig. 13 (a)), it can be found that after the transfer

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process from furnace to water, the surface temperature of the thin sheet is about 513 ℃ which is still appropriate for quenching of 2219 aluminum alloy (Tiryakioğlu and

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Shuey, 2010). In the whole immersing process, the temperature of the part exposed in the air is still kept at 513 ℃, while the part immersed into water is cooled rapidly. Based on the simulation results, a schematic diagram of temperature distribution

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is plotted, as shown Fig. 13 (b). Here, it is assumed that the temperature distribution is uniform along width (x) and thickness (z) directions. Along length (y) direction, the

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immersed part with non-uniform temperature distribution is divided into some small

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isothermal zones (1, 2…n) where T1 ＜ T2 ＜…＜ Tn , and the exposed part is kept at the uniform temperature of T0 . During quenching, these small isothermal zones is cooled and shrinks. This shrinkage causes these zone subjected an edge displacement along x direction and edge displacement ux , as shown in Fig. 13 (b). Thus, the onset of the quenching distortion can be considered as a structural instability under the action of edge displacement. The edge displacement uxn for the n zone can be calculated as： 21 / 61

uxn   n1l Tn  Tn1   uxn1

(9)

where,  n 1 , l and uxn 1 indicate the thermal expansion coefficient at Tn 1 , width of thin sheet, and x edge displacement for the n-1 zone, respectively. Thus, the edge displacement ux of the zone near the water surface can be expressed as Eq. (10). It can be known from Eq. (10) that ux is determined by the temperature gradient at every position of the immersed part, i.e., a large temperature gradient means a large displacement. Eq. (9) shows that the displacement of an

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isothermal zone is the accumulation of the displacements below this isothermal zone, which means that the zone near the water surface has the largest displacement (Eq. (10)) and will be bent first.

ux   nl T0  Tn    n 1l Tn  Tn 1    n  2l Tn 1  Tn  2   ......  1l T2  T1 

re

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 T  T  T  T  T  T  T  T    yl  n 0 n   n 1 n n 1   n 2 n 1 n  2  ......  1 2 1  (10) y y y y    yl  n grad Tn   n 1 grad Tn 1  ......  1 grad T1 

where, y and grad Tn indicate the width of isothermal zones and temperature

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gradient along y direction at Tn , respectively.

Fig. 13. (a) Temperature distribution of thin sheet in quenching and (b) corresponding schematic illustration.

On the other hand, there will be compression stress along x direction (σx) corresponding to edge displacement ux . Fig. 14 shows the distributions of σx. Here, 22 / 61

define the surface that the quenching distortion begins as the concave surface, while the opposite surface as the convex surface. Since the stress distributes symmetrically along the central axis, the stress contour of concave and convex surfaces are displayed in the left and right side of Fig. 14, respectively. It can be seen that the edge displacement ux that is contributed by immersed part leads to the compressive stress concentration around the exposed zone near the water surface. The compression decreases with the immersing speed, while it increases with the immersion depth. The increasing compression results in the

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distortion instability of thin sheet. Thus, the exposed zone near the water surface can be considered as the distortion zone, and the distortion of the exposed part is caused by this zone.

Before the distortion instability occurring, the concave and convex surfaces show

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the same stress distribution (Fig. 14 (a)-(c)), while they show different stress

distributions after the occurrence of distortion instability (Fig. 14 (d)-(f)). Because the

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compression increases with the decrease of immersing speed, higher immersion depth is needed to initiate the distortion for the faster immersing speed. Thus, it can be

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observed that the critical immersion depth, namely the depth as quenching distortion

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occurs, decreases with the decrease of immersing speed.

Fig. 14. σx variation on the surface with different immersing speeds and immersion depths: (a) 0.32 m/s, 10%; (b) 0.10 m/s, 10%; (c) 0.05 m/s, 10%; (d) 0.32 m/s, 44%; (e) 0.10 m/s, 20%; (f) 0.05 m/s, 13%. 23 / 61

5.2.2 Growth of distortion mode In this section, to describe the formation of distortion mode of thin sheets, the bending modes of different cross-sections in width were analyzed. Based on the results of 5.2.1, it can be known that the distortion of thin sheet is triggered by the bending of distortion zone. The bending degree and direction of the upper and bottom edges determine the distortion mode of thin sheets. Therefore, in order to know how the distortion mode of thin sheet forms, it is necessary to investigate the relationship

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between the bending modes of the upper edge, bottom edge and the distortion zone. Figs. 15-17 show the bending modes of the upper edge, bottom edge and the distortion zone under different immersing speeds and immersion depths, and the

contour maps in these figures indicate the displacement along the bending direction.

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The bending heights have been magnified 5 times to clearly display the bending

modes of the distortion zone. At the immersing speed of 0.32 m/s (Fig. 15), the mode

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of bending distortion zone is a one-half sine wave (mode 1), and the upper edge and bottom edge have the same bending mode as the distortion zone with a small bending

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height when the instability occurs (Fig. 15 (a)). As the immersion depth increases, the bending height of the distortion zone and two edges becomes larger while the mode

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and bending direction do not change (Fig. 15 (b) and (c)). Thus, in this case, the thin sheet exhibits the first distortion mode in which the upper and bottom edges bend in

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the same direction.

Fig. 15. Bending modes of upper edges, distortion zones and bottom edges under immersing speed of 0.32 m/s at different immersion depths: (a) 44%, (b) 49% and (c) 54%. 24 / 61

At the immersing speed of 0.10 m/s, distortion zone, upper edge and bottom edge are all presented mode 1 with the same bending direction at the critical immersion depth 20% (Fig. 16 (a)). As the immersion depth increases to 25%, however, the mode 1 in distortion zone seems to be suppressed (Fig. 16 (b)), and the side regions become two 1/2 half waves (denoted by the rectangles). Thus, at this moment, the distortion zone shows a two-half sine wave shape (mode 2). When the immersion depth increases to 30%, two new half waves were observed (denoted by the circles), and the

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distortion zone shows a three-half sine wave shape (mode 3) (Fig. 16 (c)). With the formation of mode 3, the upper edge bends to the opposite direction of the bottom edge (Fig. 16 (c)), and thus the quenched thin sheet exhibits the second distortion

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mode.

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Fig. 16. Bending modes of upper edges, distortion zones and bottom edges under immersing speed of 0.10 m/s at different immersion depths: (a) 20%, (b) 25% and (c) 30%.

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At the immersing speed of 0.05 m/s, the upper edge, distortion zone and bottom edge show mode 1 with the same bending direction at the critical immersion depth 13%

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(Fig. 17 (a)). With the increase of the immersion depth, the bending directions of these regions do not change while the bending height is larger (Fig. 17 (b)). However, when the immersion depth increases to 85%, the bending heights in these regions become rather small, which means bending behavior is strongly suppressed in this case (Fig. 17 (c)).

25 / 61

Fig. 17. Bending modes of upper edges, distortion zones and bottom edges under immersing speed of 0.05 m/s at different immersion depths: (a) 13%, (b) 36% and (c) 85%.

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Based on the analyses above, it can be concluded that the distortion modes of the quenched thin sheets are significantly affected by the bending modes in distortion zones. Therefore, it is necessary to investigate the formation conditions of the

different bending modes in this zone. For the purpose, the bending behavior of the

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distortion zone is analyzed through buckling theory (Cao and Boyce, 1997b).

As well known, one-half sine wave buckling will be occur readily if a sheet is

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compressed by edge compression and there is no lateral constraint (Timoshenko, 1961). However, if the mode 1 is suppressed due to the constraint, then mode 2 will

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be favored and so on (Cao and Boyce, 1997b). Kim et al. (2000) studied the bending behavior of strips subjected to a compression under a constraining wall. They found

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that when the strip was obstructed, the mode of buckling transits from mode 1 to mode 2 to mode 3 with the increase of engineering strain. Therefore, it can be considered that the strong lateral constraints are the requirements for the formation of

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higher buckling mode.

During quenching process, the thin sheets distort freely, thus the constraints

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imposed on distortion zone are not from the external force but the uncoordinated distortion in different parts of the quenching sheets. In this section, in order to know which part of the thin sheet provides the constraint and explain the reason behind the variation of bending mode in distortion zone during quenching process, the temperature gradient distributions along the y direction (the path denoted in Fig 13 (a)) at the immersion depths of 20%, 50% and 100% were analyzed, as shown in Fig. 18. 26 / 61

It can be observed that, when the temperature exceeds 100 ℃, i.e., above the red

dot line, the temperature gradient (the slope of the curves) along y direction on the immersed part of the thin sheet is obviously larger than that when the temperature is below 100 ℃ at the same immersion depth. Thus, based on Eq. (10), it can be known that the immersed part above 100 ℃ has the main contribution to the edge displacement of distortion zone, and is defined as active distortion zone (ADZ), as denoted by the green arrows in Fig. 18. For the part with temperatures lower than 100 ℃, its contribution to the bending of the distortion zone is very small because of

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the obvious decrease of the temperature gradient. Meanwhile, the x displacement of this part is significantly reduced, thus it has a tendency to stretch under the action of elastic force. This elastic stretch can provide a constraining pressure to suppress the

bending of the distortion zone. Therefore, the part with temperatures lower than 100 ℃

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is defined as the constraint zone (CZ). For instance, consider the case of 100%

immersion depth in Fig. 18, the zone blow the intersection of the red dot line and the

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dash curve is the CZ, i.e., the yellow shaded zone showing in the figure.

Fig. 18. Temperature profiles at the immersion depths of 20%, 50% and 100% and at different 27 / 61

immersing speeds: (a) 0.05 m/s, (b) 0.10 m/s and (c) 0.32 m/s.

The temperature gradient distribution determines the edge displacement ux , and the constraining pressure of the CZ increases with increasing its area and decreasing its temperature. It can be observed from Fig. 18 that, at a certain immersing speed, after the area of ADZ reaches the maximum value, the temperature gradient distribution and area of ADZ does not change with increasing the immersion depth (such as at 20%, 50% and 100%), while the area of the CZ keeps increasing.

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Therefore, the edge displacement ux in distortion zone would barely change when it reaches a maximum, while the constraining pressure increases rapidly after this

moment. Moreover, with decreasing the immersing speed, the area (the yellow shaded zone showing in Fig. 18) increases and temperature of the CZ decreases at the same

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immersion depth, that is, the maximum constraining pressure imposed by the CZ in

immersing process increases. In addition, in the case of relatively slower immersing

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speed, it can be observed from Fig. 15-17 that the maximum bending height of the distortion zone is lower, thus, the corresponding maximum edge displacement ux is

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smaller. Based on the above analysis, three possible constraining pressure variation trajectories under various immersing speed are given to describe this change rule, as

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denoted by the red arrow in Fig. 19. The uxA at point A indicates the edge displacement needed to initiate buckling in the distortion zone.

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To determine the relationship between lateral constraint and the buckling mode for a rectangular plate, a criterion was given by Cao and Boyce (Cao and Boyce,

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1997b), as shown in Fig. 19. The figure shows that the maximum constraining pressures needed to suppress buckling are functions of the edge displacement for the mode n (n=1, 2, 3) buckling (Cao and Boyce, 1997b). In Fig. 19, the three black curves cross over each other at the transition pressures P12 and P23. The P12 and P23 are the constraining pressures where the favored mode of buckling transits from mode 1 to mode 2 and from mode 2 to mode 3, respectively. This criterion can be used to explain the variation of buckling mode in the distortion zone of the quenching thin 28 / 61

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sheet.

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Fig. 19. Maximum constraining pressures as functions of the edge displacement for the mode 1, 2 and 3 buckling (black curves), and the variation trajectories of constraining pressure imposed on distortion zone under various immersing speed (red arrows).

As shown in Fig. 19, at the fast immersing speed (such as 0.32 m/s), the

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trajectory (denoted by the red dot arrow) only enters ABFG area (mode 1 area), in which the distortion zone would favor mode 1 because the critical edge displacement needed to initiate buckling for mode 1 is lower than that for mode 2 and mode 3.

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Therefore, in the case of fast immersing speed, only mode 1 is observed in the distortion zone (Fig. 15).

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As the immersing speed decreases to the 0.10 m/s, the trajectory (denoted by the red short dash arrow) enters mode 1 area first, and then crosses the BF line and CE

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line into the BCEF area (mode 2 area) and CDE area (mode 3 area) in turn when the constraining pressure exceeds the critical pressures P12 and P23, respectively. In the mode 2 area, because the critical edge displacement for mode 2 is lower, the transition of buckling mode in distortion zone from mode 1 to mode 2 is observed (Fig. 16 (b)). Similarly, in the mode 3 area, the buckling mode transits from mode 2 to mode 3, which causes the upper edge to be bent to the opposite direction of the bottom edge (Fig. 16 (c)). 29 / 61

With further decrease of immersing speed to the slow speed (such as 0.05 m/s), the trajectory (denoted by the red dash arrow) enters mode 1 area first and then may enter mode 2 and mode 3 area, thus the distortion zone could buckle at the early stage of quenching. However, as the constraining pressure increases, the trajectory passes through the curves AB, BC or CD (most likely AB for our study) and enters the upper left area in which the buckling of the distortion zone always be suppressed. Therefore, at slow immersing speed, the distortion zone almost becomes straight with increasing the immersion depth (Fig. 17).

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The mode variations of the bending distortion zone significantly affect the bending degree and direction of the upper and bottom edges, and subsequently determine the distortion mode of thin sheets. From the above analyses, the

relationship between bending mode of the distortion zone and the distortion mode of

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the thin sheet can be concluded as follows. when the mode of bending distortion zone is one-half sine wave, the thin sheet exhibits the first initial distortion mode that the

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upper edge and bottom edge have the same bending direction, and eventually grows to be a saddle-shaped (Fig. 20). When the bending mode is three-half sine wave, the

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snap-through occurs and the thin sheet exhibits the second initial distortion mode that the upper edge and bottom edge have opposite bending direction, and showing

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shovel-shaped distortion mode finally. When the bending of distortion zone is suppressed, the thin sheet exhibits the third initial distortion mode where both of the upper and bottom edges present quite small bending height, and eventually become an

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arch-shaped distortion mode.

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6. Conclusions

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Fig. 20. Schematic illustration of the quenching distortion process.

The distortion behavior and its mechanism of 2219 aluminum alloy thin sheets in

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quenching process were investigated based on the experiments and finite element analysis considering the immersing process. The main conclusions are drawn as

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follows:

1) The in-situ observations and simulations show that the quenching thin sheets

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present saddle-shape in the early stage of quenching at all experimental immersing speeds, and then keep the saddle-shape for the relatively fast immersing speed, while exhibit shovel-shape after snapping occurs and arch-shape after bending in width

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recovers for the relatively slow immersing speeds. 2) According to the characteristics of temperature and stress distributions

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obtained from simulation, the quenching thin sheet is divided into the constraint zone, the active distortion zone and the distortion zone. The distortion zone indicates the exposed part with maximum bending distortion. The active distortion zone with large temperature gradient and constraint zone with small temperature gradient can promote and obstruct the bending of distortion zone, respectively. 3) With the combined effect of the active distortion zone and constraint zone, the distortion zones show different bending modes which determine the final distortion 31 / 61

modes of the quenched thin sheets. When the bending mode is one-half sine wave, the quenching thin sheets always show saddle-shape. For the cases where the bending mode changes to three-half sine wave and the bending of distortion zone is suppressed, the quenching thin sheets show shovel-shaped and arch-shaped distortion modes, respectively. 4) The bending behavior of the distortion zone is analyzed based on a buckling criterion under laterally constrained conditions. In quenching process, with the decrease of immersing speed, the constraining pressure from constraint zone increases,

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which results in the variation of the bending mode of the distortion zone in quenching from one-half sine wave to three-half sine wave to straight.

Author Contributions Section

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Z.X. Li: Conceptualization; Methodology; Investigation; Writing - Original Draft

M. Zhan: Writing - Review & Editing; Supervision; Project administration; Funding

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acquisition X.X. Wang: Investigation F. Ma: Investigation R. Li: Investigation

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X.G. Fan: Conceptualization; Writing - Review & Editing; Funding acquisition

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Declaration of Interest Statement The authors declare that they have no known competing financial interests or

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paper.

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personal relationships that could have appeared to influence the work reported in this

Acknowledgements The authors would like to acknowledge the support from the National Science

Fund for Distinguished Young Scholars of China (Project 51625505), Key Program Project of the Joint Fund of Astronomy and National Natural Science Foundation of China (Project U1537203), and the Research Fund of the State Key Laboratory of Solidification Processing (Projects 118-TZ-2015). 32 / 61

References ABAQUS, 2014. Abaqus analysis user’s guide (version 6.14). Cao, J., Boyce, M.C., 1997a. A predictive tool for delaying wrinkling and tearing failures in sheet metal forming. Trans. ASME, J. Engng Mater. Technol. 119, 354-365. Cao, J., Boyce, M.C., 1997b. Wrinkling behavior of rectangular plates under lateral constraint. Int. J. Solids. Struct. 34, 153-176. Dursun, T., Soutis, C., 2014. Recent developments in advanced aircraft aluminium

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alloys. Mater. Des. 56, 862-871. Gusic, G., Combescure, A., Jullien, J.F., 2000. The influence of circumferential thickness variations on the buckling of cylindrical shells under external compression. Comput. Struct. 74, 461-477.

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Khan, S., Hussain, G., Ilyas, M., Rashid, H., Khan, M.I., Khan, W.A., 2018.

Appropriate heat treatment and incremental forming route to produce age-hardened

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components of al-2219 alloy with minimized form error and high formability. J. Mater. Process. Technol. 256, 262-273.

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Kim, J.B., Yoon, J.W., Yang D.Y., 2000. Wrinkling initiation and growth in modified Yoshida buckling test: Finite element analysis and experimental comparison. Int. J.

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Mech. Sci. 42, 1683-1714.

Koç, M., Culp, J., Altan, T., 2006. Prediction of residual stresses in quenched aluminum blocks and their reduction through cold working processes. J. Mater.

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Process. Technol. 174, 342-354.

Li, J., Carsley, J.E., Stoughton, T.B., Hector, L.G. Jr., Hu, S.J., 2013. Forming limit

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analysis for two-stage forming of 5182-O aluminum sheet with intermediate annealing. Int. J. Plast. 45, 21-43.

Liu, N., Yang, H., Li, H., Li, Z.J., 2014. A hybrid method for accurate prediction of multiple instability modes in in-plane roll-bending of strip. J. Mater. Process. Technol. 214, 1173-1189.

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Li, Y.N., Zhang, Y.A., Li, X.W., Li, Z.H., Wang, G.J., Jin, L.B., Huang, S.H., Xiong, B.Q., 2016. Quenching residual stress distributions in aluminum alloy plates with different dimensions. Rare Met. 1-11. Li, Z.X., Zhan, M., Fan, X.G. Ma, F., Wang, J.W., 2018. Constitutive model over wide temperature range and considering negative-to-positive strain rate sensitivity for as-quenched AA2219 sheet. J. Mater. Eng. Perform. 28, 1-10. Maniruzzaman, M., Dai, M.X., Sisson, R.D., 2012. Effect of quench start temperature on surface heat transfer coefficients. In: MacKenzie DS (Ed) Quenching control

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and distortion 2012: Proceedings of the 6th International Quenching and Control of Distortion Conference Including the 4th International Distortion Engineering Conference, Chicago, Illinois, USA, pp. 57–68.

process simulation. J. Phys. IV 120, 269–276.

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Maniruzzaman, M., Sisson, R.D., 2004. Heat transfer coefficients for quenching

Nallathambi, A.K., Kaymak, Y., Specht, E., Bertram, A., 2008. Distortion and

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residual stresses during metal quenching process. In: Bertram, A., Thomas, J. (Eds.), Micro-Macro-Interactions in Structured Media and Particle Systems. Springer,

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Berlin Heidelberg, pp. 145–157.

Narazaki, M., Kogawara, M., Shirayori, A., Kim, S.Y., Kubota, S., 2012.

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Experimental and simulation studies on asymmetrical quench distortion of long thin steel parts. International Quenching and Control of Distortion Conference American Society for Metals. 28, 3369-3382.

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Papadopoulos, V., Papadrakakis, M., 2005. The effect of material and thickness variability on the buckling load of shells with random initial imperfections. Comput.

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Methods Appl. Mech. Eng. 194, 1405-1426.

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Sugianto, A., Narazaki, M., Kogawara, M., Shirayori, A., 2008. A comparative study on determination method of heat transfer coefficient using inverse heat transfer and iterative modification. J. Mater. Process. Technol. 209, 4627-4632 34 / 61

Silva, A.D.D., Pedrosa, T.A., Gonzalez-Mendez, J.L., Jiang, X., Cetlin, P.R., Altan, T., 2012. Distortion in quenching an AISI 4140 C-ring-predictions and experiments. Mater. Des. 42, 55-61. Totten, G.E., Mackenzie, D.S., 2000. Aluminum quenching technology: a review. Mater. Sci. Forum. 331, 589-594. Timoshenko, S., 1961. Theory of elastic stability. McGraw-Hill, New York. Tiryakioğlu, M., Shuey, R.T., 2010. Quench sensitivity of 2219-T87 aluminum alloy plate. Mater. Sci. Eng. A 527, 5033-5037.

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Wang, H., Yi, Y., Huang, S., 2016. Influence of pre-deformation and subsequent ageing on the hardening behavior and microstructure of 2219 aluminum alloy forgings. J. Alloys Comp. 685, 941-948.

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Wong, W., Pellegrino, S., 2006. Wrinkled membranes III: numerical simulations. J.

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[Figure captions] Fig. 1. Quenching experiments: (a) initial shape of thin sheet and (b) schematic illustration of quenching process. Fig. 2. Measurement of HTC: (a) illustration of sheet probe, (b) measured water cooling curve and inversely determined HTC curve. Fig. 3. Thickness distribution of the initial sheet. Fig. 4. Meshed geometry used for the simulation. Fig. 5. Flowchart for quenching analysis.

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Fig. 6. Measurement of BH: (a) the bottom, upper and side edges of thin sheet, and (b) positive BH and (c) negative BH for bottom edge and side edge, respectively. Fig. 7. Images of thin sheets after quenching.

(t/a).

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Fig. 8. Distortion modes of quenched thin sheets with different thicknesses ratios

Fig. 9. Comparison of BHs between predicted results and experimental results: (a)

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bottom edges, (b) upper edges and (c) side edges.

Fig. 10. Shape variation of thin sheet under immersing speed of 0.32 m/s with

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immersion depth: (a) 30%, (b) 50%, (c) 80% and (d) 100%. Fig. 11. Shape variation of thin sheet under immersing speed of 0.10 m/s with

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immersion depth: (a) 30%, (b) 50%, (c) 60% and (d) 100%. Fig. 12. Shape variation of thin sheet under immersing speed of 0.05 m/s with immersion depth: (a) 40%, (b) 55%, (c) 80% and (d) 100%.

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Fig. 13. (a) Temperature distribution of thin sheet in quenching and (b) corresponding schematic illustration.

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Fig. 14. σx variation on the surface with different immersing speeds and immersion depths: (a) 0.32 m/s, 10%; (b) 0.10 m/s, 10%; (c) 0.05 m/s, 10%; (d) 0.32 m/s, 44%; (e) 0.10 m/s, 20%; (f) 0.05 m/s, 13%. Fig. 15. Bending modes of upper edges, distortion zones and bottom edges under immersing speed of 0.32 m/s at different immersion depths: (a) 44%, (b) 49% and (c) 54%. Fig. 16. Bending modes of upper edges, distortion zones and bottom edges under 36 / 61

immersing speed of 0.10 m/s at different immersion depths: (a) 20%, (b) 25% and (c) 30%. Fig. 17. Bending modes of upper edges, distortion zones and bottom edges under immersing speed of 0.05 m/s at different immersion depths: (a) 13%, (b) 36% and (c) 85%. Fig. 18. Temperature profiles at different immersing speeds: (a) 0.05 m/s, (b) 0.10 m/s and (c) 0.32 m/s. Fig. 19. Maximum constraining pressures as functions of the edge displacement for

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the mode 1, 2 and 3 buckling (black curves), and the variation trajectories of constraining pressure imposed on distortion zone under various immersing speed (red arrows).

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Fig. 20. Schematic illustration of the quenching distortion process.

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Figure 9 18

(a)

6 0

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BH (mm)

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0

40 80 120 160 Botton edges-distance/mm

200

24 (b) 18 FEM simulation-0.32m/s Exp.-0.32m/s FEM simulation-0.10m/s Exp.-0.10m/s 12 FEM simulation-0.05m/s Exp.-0.05m/s 6 0 -6 -12 -18 -24 0 40 80 120 160 200 Upper edges-distance (mm) 18

(c)

12

Exp.-0.32m/s Exp.-0.10m/s Exp.-0.05m/s

6

FEM simulation-0.32m/s FEM simulation-0.10m/s FEM simulation-0.05m/s

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BH (mm)

FEM simulation-0.32m/s FEM simulation-0.10m/s FEM simulation-0.05m/s

0 -6

0

Immersing direction 40 80 120 160 Side edges-distance (mm)

200

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Exp.-0.32m/s Exp.-0.10m/s Exp.-0.05m/s

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BH (mm)

12

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Jo

ur

na

lP

re

-p

ro of

Figure 10

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Jo

ur

na

lP

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-p

ro of

Figure 11

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Jo

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na

lP

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Figure 12

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Jo

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na

lP

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-p

ro of

Figure 13

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Jo

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na

lP

re

-p

ro of

Figure 14

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Jo

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na

lP

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-p

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Figure 15

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Jo

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na

lP

re

-p

ro of

Figure 16

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Jo

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na

lP

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-p

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Figure 17

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Jo

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na

lP

re

-p

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Figure 18

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Jo

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lP

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-p

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Figure 19

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Jo

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lP

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Figure 20

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[Table captions] Table 1 Main chemical compositions of 2219 aluminum alloy. Table 2 Main thermal parameters of 2219 aluminum alloy at different temperatures.

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Table 3 Elastic modulus and yield strength at different temperatures.

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Table 1 Cu

Mn

Fe

Zr

Ti

Si

Zn

Mg

Al

wt.%

6.5

0.36

0.21

0.18

0.06

0.05

0.02

0.01

Bal.

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Element

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Table 2 Thermal expansion (10-6K-1)

Thermal conductivity (W/m·k)

Specific heat (J/g/K)

22.707 24.246 25.989 26.587 26.604 26.116

103.793 109.553 122.849 136.363 138.533 122.685

1.137 1.151 1.197 1.263 1.297 1.345

T (℃)

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na

lP

re

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ro of

25 100 200 300 400 530

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Table 3

65.891

58.518

44.973

0.001 0.01 0.1 0.001 0.01 0.1 0.001 0.01 0.1 0.001 0.01 0.1 0.001 0.01 0.1 0.001 0.01 0.1

Jo

ur

na

lP

28.652

172 165 158 160 152 151 128 136 141 109 128 141 58 73 88 7 11 19

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T (℃)

25

100

200

ro of

72.752

Strain rate (s-1)

-p

75.890

Yield strength (MPa)

re

Elastic modulus (GPa)

300

400

530