tension leveling using finite element analysis with
TRANSCRIPT
ISIJ International, Vol. 60 (2020), No. 6
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ISIJ International, Vol. 60 (2020), No. 6, pp. 1273–1283
* Corresponding author: E-mail: [email protected]: https://doi.org/10.2355/isijinternational.ISIJINT-2019-620
1. Introduction
With the increasing demand for high-quality metal sheets, tension leveling is being increasingly applied in metal strip production lines to ensure the flatness of the produced metal strips. In tension leveling, the leveled strip is subjected to cyclic deformations under a combination of tension and bending to remove curvature and waviness defects. For the development of leveling technology, since finite ele-ment (FE) analysis is an effective means of investigating the relationship between process parameters and the final production quality, it has been used to assist the process design of tension leveling. Yoshida and Urabe developed a two-dimensional (2D) FE code for tension leveling and provided an appropriate leveler roll arrangement.1) Huh et al. conducted a simulation-based process design for tension leveling using 2D FE analysis to investigate the relation-ship between the leveler roll intermesh and the curl level-ing effect.2) Morris et al. conducted a parametric study of the tension leveling process using the Taguchi method and a commercial FE code to investigate the effects of the roll wrap angle, the line speed, the total elongation and the yield stress on the residual flatness after tension leveling.3)
Although the FE analysis of tension leveling with a clas-sical constitutive model under plane stress and strain condi-tions can reflect the trend of experimental results and has acceptable accuracy, many advanced constitutive models for materials have been applied in other forming processes to give a better prediction for FE analysis to reduce the
Tension Leveling Using Finite Element Analysis with Different Constitutive Relations
Honghao WANG,1) Boxun WU,2)* Takuya HIGUCHI1) and Jun YANAGIMOTO1)
1) Graduate School of Engineering, The University of Tokyo, Hongo 7-3-1, Bunkyo, Tokyo, 113-8656 Japan.2) Institute of Industrial Science, The University of Tokyo, Komaba 4-6-1, Meguro, Tokyo, 153-8505 Japan.
(Received on September 27, 2019; accepted on December 25, 2019; J-STAGE Advance published date: February 28, 2020)
Tension leveling is applied in metal strip production lines to improve the flatness of metal strips by a combination of tension and bending. To develop tension leveling technology, finite element (FE) analysis is increasingly used in tension leveling process design to reduce the number of trial productions and pro-vide a deeper insight into the process. For the FE analysis of tension leveling, since the material properties affect the leveling results, a material constitutive model that can accurately describe the material behaviors during tension leveling should be applied. In our previous investigation, an advanced constitutive model was constructed for the FE analysis of tension leveling with high accuracy. Here, we report the results of an analysis on tension leveling to clarify the effect of constitutive relations on FE analysis results. Leveling mechanisms for high-strength steel strips were also clarified on the basis of FE analysis results.
KEY WORDS: tension leveling; finite element analysis; constitutive relation.
number of trial productions in process design. Kim et al. conducted the limiting dome height test and accurately predicted the sheet formability using FE analysis with the Hill48 and Yld2000-2d yield functions.4) Sumikawa et al. improved the springback prediction accuracy by using the Yld2000-2d yield criterion and the Yoshida–Uemori model in the FE analysis of the press forming test.5,6) Wu et al. obtained the FE analysis results of a deep drawing test with high accuracy by using the Hill48 yield criterion with the non-associated flow rule.7) Wu et al. developed a non-asso-ciated constitutive model based on the Yld91yield function and the Hill48 plastic potential function with an anisotropic hardening model to conduct the FE analysis of a circular hole expansion test accurately.8) Since process parameters such as the leveler roll intermesh and material properties such as the yield stress and the Bauschinger effect have a significant effect on the final quality of the leveled strip, when conducting leveling process design using FE analysis, it is also necessary to apply an advanced constitutive model to describe the material properties accurately during tension leveling.9,10) In our previous investigation, an advanced con-stitutive model, which can describe the anisotropy and the Bauschinger effect under three-dimensional (3D) strain and stress conditions for materials, was constructed and applied to achieve the FE analysis of tension leveling with high accuracy.11) However, how the constitutive relations (e.g., 3D stress and strain conditions, material anisotropy, and work hardening behaviors) affect the analysis results has not been clarified in previous studies. In this investigation, aiming at delivering basic information for the process design of tension leveling, FE analyses of the FE model dimension, the yield criterion with the flow rule and the hardening rule,
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together with theoretical analysis, were conducted to clarify the effect of the constitutive relations on FE analysis results. Moreover, the leveling mechanisms for an SPFC980 strip were also clarified on the basis of FE analysis results.
2. Material Behavior Modeling and Implementation in FE Analysis
2.1. Material Constitutive ModelIn tension leveling, since strip defects are in different
directions and the strip is subjected to cyclic loading, it is necessary to consider the material anisotropy and the Bauschinger effect under 3D stress and strain conditions during the modeling of the material behavior to describe the material behaviors in tension leveling with high accu-racy. In this investigation, the Hill48 yield criterion with the non-associated flow rule and the mixed hardening rule were adopted to construct the material constitutive model.11) The Hill48 yield criterion with the non-associated flow rule is given by
f
F G H L M
Y
yy zz zz xx xx yy yz
��� � � � �
��� � � �� � � �� � � �
� �
� � � � � � �3
2
2 22 2 2 2 �� �zx xyN
F G H
2 22�
� �
.......................................... (1)
dg
d� �P �� � ��
����
............................ (2)
g
F G H L Myy zz zz xx xx yy yz
�
� � � � � � �
� � �
��� � � �� � � �� � � �3
2
2 22 2 2 2* * * * *�� �zx xyN
F G H
2 22�
� �
*
* * *
.......................................... (3)
where f(η) and g(η) are the yield function and plastic potential function, respectively. F, G, H, L, M and N are the parameters of the yield function. F*, G*, H*, L*, M* and N* are the parameters of the plastic potential function. η repre-sents the stress state on the yield surface centered on the back stress, where η=σ−α, with σ and α denoting the Cauchy stress tensor and back stress tensor, respectively. Here, subscripts x, y, and z are taken as the longitudinal direction, width direction and thickness direction, respectively. dλ is the plastic multiplier. η , ε p and σ Y are the equivalent stress, equivalent plastic strain and yield stress, respectively. The material parameters of the Hill48 anisotropic yield criterion with the non-associated flow rule are given as
21 1 1
902 2
02
Fb
� � �� � �
...................... (4-a)
21 1 12
02
902
Gb
� � �� � �
...................... (4-b)
21 1 1
02
902 2
Hb
� � �� � �
...................... (4-c)
2 2 212
L M Ns
� � ��
...................... (4-d)
Fr
r r* �
�� �0
90 01 .......................... (5-a)
Gr
* ��1
1 0
.............................. (5-b)
Hr
r* �
�0
01 ............................. (5-c)
L M N
r r r
r r* * *� � �
� � �� ��� �
�0 90 45
90 0
1 2
2 1 ............ (5-d)
where σ0, σ90, σb, τs, r0, r45 and r90 are the yield stresses and Lankford values measured from the tensile, shear and bulge tests.7) The material anisotropy in both the stresses and strains can be described by applying the Hill48 yield criterion with the non-associated flow rule. The mixed hard-ening rule is given by
� ��
� � �Y bQ eC
ep p
� � �� � � �� �� �0 1
1
1 1 1 ........... (6)
�� �� ��� �1 2 ............................. (7-a)
d
C
gd p��
����
��1 1�� �
��
���
�
���
11� � .................. (7-b)
dC
gd p��
����
2 � � �2 � .......................... (7-c)
where the back stress tensor α is divided into two parts, α1 and α2. σ0, Q, b, C1, γ1 and C2 are the parameters for the mixed hardening rule. By conducting a tensile-compressive test, the parameters for mixed hardening can be identified from the strain–stress curve. Zang et al. proposed this mixed hardening rule to describe material behaviors during cyclic loading such as the Bauschinger effect and the transient behavior.12)
2.2. Finite Element ImplementationThe constructed constitutive model for the FE analysis
of tension leveling under 3D stress and strain conditions was implemented using the user-defined material subroutine with the return-mapping algorithm, which is an implicit integration scheme for the constitutive model (details in reference 11)). To obtain the strain and stress fields with high accuracy, the FE analyses were conducted with a static analysis step, which can reduce the oscillation of calculated results compared with the case of using a dynamic analysis step.13)
For the FE model of tension leveling, the 3D analytical rigid body was used to model the leveler rolls. For the lev-eled strips, since bending is one of the main deformation modes in tension leveling, and the second-order brick ele-ment can ensure the continuity of the bending moment at element interfaces (detailed information is given in Appen-dix I), to conduct the FE analysis of tension leveling under 3D stress and strain conditions with both high accuracy and efficiency, the 20-node second-order brick element with reduced integration was used to model the leveled strips. For the contact between the leveler roll and the leveled strip, the penalty friction formulation was applied and the fric-tion coefficient was set to 0.15. During the FE analyses, the
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angular velocity of each leveler roll was carefully examined to ensure that each leveler roll was simulated as an idle roll, and the maximum time increment of each analysis step was set to 10 −5 to control the maximum strain increment within the limit to ensure the convergence of the static analysis calculations.
2.3. Accuracy Verification of Constructed Material Constitutive Model
To verify the accuracy of the constructed material con-stitutive model, FE analysis of the tension leveling for an SPCC strip (dimensions: length × width × thickness = 1 300 mm × 200 mm × 0.5 mm) was conducted with the proposed material constitutive model in Abaqus/Standard. The arrangement of leveler rolls set by Yoshida and Urabe is shown in Fig. 1 and the material parameters of the Hill48 yield criterion with the non-associated flow rule and the mixed hardening rule for SPCC are listed in Tables 1 and 2 (non-AFR Hill48 + mixed hardening).1,11) The Young’s modulus and Poisson’s ratio of SPCC are assumed to be 200 GPa and 0.3, respectively. The applied mesh size for the second-order brick element is 5 mm × 20 mm × 0.125 mm (a case study shown in Table 3 was conducted to decide this mesh size considering both the accuracy and the efficiency of the FE analysis of tension leveling) and the applied entry tension is 40 MPa. The 3D FE analysis results of the strip residual curvature after tension leveling with different intermeshes obtained by the constructed constitutive model are compared with the experimental results of Yoshida and Urabe (the residual curvature obtained by the 3D FE analy-sis is the average value of the results of the strip center and strip edge, the residual curvature of the strip center or strip edge is calculated by three selected nodes of the central layer of the strip center or strip edge after the springback).1) Since the first-order brick element is applied in our previous
investigation for the FE analysis of tension leveling, the FE analysis results obtained by the second-order brick element in this investigation are also compared with the results obtained by the first-order brick element.11) As shown in Fig. 2, the FE analysis of tension leveling considering both the material anisotropy and material behaviors in cyclic loading such as the Bauschinger effect under 3D stress and strain conditions is with high accuracy. And the accuracy of
Fig. 1. Arrangement of leveler rolls in tension leveling. (Online version in color.)
Table 1. Parameters for material anisotropy of SPCC and SPFC980.
Constitutive model H F G L, M, N H* F* G* L*, M*, N*
SPCC
AFR von Mises 1.000 1.000 1.000 3.000 1.000 1.000 1.000 3.000
AFR Hill48-σ 1.000 0.876 1.026 2.642 1.000 0.876 1.026 2.642
AFR Hill48-r 1.000 0.383 0.495 2.178 1.000 0.383 0.495 2.178
non-AFR Hill48 1.000 0.876 1.026 2.642 1.000 0.383 0.495 2.178
SPFC980 non-AFR Hill48 1.000 0.999 1.033 2.690 1.000 1.190 1.165 3.053
Table 2. Parameters for hardening rule of SPCC and SPFC980.
Constitutive model σ0 Q b C1 γ1 C2
SPCC
Isotropic hardening 158.9 171.8 17.4 0 0 0
Mixed hardening 158.9 50.4 65.4 54 468.9 889.3 1 359.7
SPFC980 Mixed hardening 579.9 269.8 168.0 94 470.0 211.1 6 331.7
Table 3. Total CPU time and simulation error of each mesh size for SPCC.
mesh size (length × width × thickness) total CPU time simulation error
sim exp. .��5 mm × 20 mm × 0.125 mm 353 709 s 0.4164 m −1
10 mm × 20 mm × 0.125 mm 273 603 s 0.6734 m −1
2.5 mm × 20 mm × 0.125 mm 1 065 639 s 0.4641 m −1
5 mm × 20 mm × 0.167 mm 332 661 s 0.7476 m −1
5 mm × 20 mm × 0.100 mm 555 076 s 0.4234 m −1
5 mm × 10 mm × 0.125 mm 1 201 701 s 0.4064 m −1
(CPU clock speed: 3.40 GHz; Number of cores per CPU: 4; Number of CPUs: 1; Main memory: 16 GB; Operating system: Windows 10)
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FE analysis of tension leveling is improved by applying the second-order brick element to ensure the continuity of the bending moment at element interfaces.
3. Case Studies of Tension Leveling by FE Analysis
Grüber and Hirt studied the process of roller leveling and concluded that the material properties affect the leveling results.10) Since different material constitutive models can describe different material properties, case studies of tension leveling were conducted using the FE model constructed in section 2 with different material constitutive models for an SPCC strip in this section to clarify the effect of the FE model dimension, material anisotropy and work hardening behaviors such as the Bauschinger effect and work harden-ing rate on the analysis results of tension leveling.
3.1. Effect of FE Model Dimension (Strip Width) on Analysis Results
In previous studies, the 2D plane strain FE analysis of tension leveling was frequently used to assist leveling process design owing to its high calculation efficiency. However, 2D plane strain FE analysis cannot reflect the metal flow behaviors along the width of the leveled strip. To clarify the effect of the FE model dimension (strip width) on the analysis results, FE analyses were conducted for SPCC strips with widths of 200 mm, 280 mm, 360 mm and infinite width (considered as the 2D plane strain model). The mate-rial constitutive model used in this case study was the Hill48 yield criterion with the non-associated flow rule and the mixed hardening rule (non-AFR Hill48 + mixed hardening, parameters listed in Tables 1 and 2). The analysis results are shown in Fig. 3. It can be concluded from the results that the strip width affects the analysis results. When the strip width increases, the residual curvature of the leveled SPCC strip also increases. During the FE analysis of tension leveling, in addition to the bending deformation applied by the leveler rolls, the leveled strip is also stretched by the tension, which makes the material flow in the width direction. When the intermesh is 10 mm, for the strip widths of 200 mm, 280 mm and 360 mm, the width reduction ratios of the leveled strip are −0.116%, −0.101% and −0.089%, respectively. In the case of infinite width, the width reduction ratio is 0 (the same as in the 2D plane strain FE analysis), which cannot reflect the actual situation. Moreover, when the intermesh is
10 mm, for the strip widths of 200 mm, 280 mm, 360 mm and infinite width, the thickness reduction ratios of the lev-eled strip are −0.125%, −0.131%, −0.135% and −0.166%, respectively, indicating that the material flow in other direc-tions can also be affected by the material flow in the width direction. Thus, the constitutive model under 3D stress and strain conditions should be applied in the 3D FE analysis of tension leveling to obtain accurate results. The 3D FE analysis cannot be replaced by 2D plane strain analysis since the material flow in the width direction (the reduction in strip width), which is important in the strip defect tension leveling such as gutter and waviness defect tension leveling, is only considered accurately in 3D FE analysis.
3.2. Effect of Material Anisotropy on Analysis ResultsThe tension leveling FE analyses of SPCC with four
different yield criteria with the associated flow rule or non-associated flow rule were conducted to clarify the effect of material anisotropy on analysis results. The parameters of Hill48 with the associated flow rule can be identified from the stresses (Eq. (4)) or r-values (Eq. (5)), which can describe the material anisotropy of stresses or strains, respectively (AFR Hill48-σ or AFR Hill48-r, parameters listed in Table 1). Here, the width of the strip is 200 mm and the thickness is 0.5 mm. When the Hill48 yield criterion is applied, the material anisotropy can be described. When the von Mises yield criterion with the associated flow rule is applied (AFR von Mises, parameters listed in Table 1), the material is isotropic. The mixed hardening rule was used in these FE analyses (parameters listed in Table 2). The analy-sis results of residual curvatures together with the experi-mental results of Yoshida and Urabe are shown in Fig. 4.1) Since the cyclic material behaviors such as the Bauschinger effect are considered by applying the mixed hardening rule, the prediction accuracy of all four models is high. It can be seen that the material anisotropy also affects the analysis results. By applying the Hill48 yield criterion with the non-associated flow rule to take the material anisotropy of both stresses and strains for SPCC into consideration, analysis results with the highest accuracy among the discussed mod-els can be obtained.
To clarify the effect of each anisotropy parameter on the analysis results, a case study of the anisotropy parameters
Fig. 3. FE analysis results of tension leveling for SPCC with dif-ferent strip widths. (Online version in color.)
Fig. 2. Comparison of experimental and FE analysis results. (Online version in color.)
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was conducted with an intermesh of 10 mm. In this case study, the Hill48 yield criterion with the associated flow rule and the mixed hardening rule was applied in the FE analyses. By adjusting the value of each anisotropy param-eter in the applied constitutive model from the isotropic situation (H=F=G=1.0, L=M=N=3.0) to the anisotropic situation, the yield function changes from the von Mises to the Hill48, and the predicted residual curvatures are shown in Fig. 5. It can be seen that H, G and F affect the analysis results. Since the anisotropic parameter G is multiplied by the stress components in the longitudinal direction and thickness directions in Eq. (1), and the values of the stress components in the longitudinal direction and thickness directions during tension leveling are larger than those of the other stress components, the variation in the anisotropic parameter G has the most significant effect on the residual curvature of the strip after tension leveling. N, which is related to the in-plane yield shear stress σxy, has little effect on the analysis results.
Chakrabarty et al. studied the bending process for aniso-tropic sheet metal and found that the material anisotropy affects the bending process.14) Since tension leveling is a combination of bending and stretching, the analysis results of tension leveling are also affected by the material anisot-ropy. In the bending process for a strip considering the anisotropy, the curvature of the central layer and the bending moment at the beginning of plastic deformation are respec-tively given by Eqs. (8) and (9) according to classical bend-ing theory (a detailed derivation is given in Appendix II),
�� � �
�cy
yn
hE�
� �� �� �� �
1 2
1
2
........................ (8)
M zdA zwdzn why
xxA
h
xx
y
h
h� � � �� ��� �
�2
3
2
....... (9)
where
n
F G H
G H� �
� �� � �� ��� � �� �
2
3
1
1 2
2
2
�
� ................ (10)
v, E, w, h and σ y are Poisson’s ratio, Young’s modulus,
the strip width, half of the strip thickness and the yield stress, respectively. It can be seen from Eq. (10) that only F, G and H affect the calculated results. Since in classical bending theory, shear deformation is not considered, N has no effect on the calculated results, which coincides with the FE analysis results of tension leveling.
3.3. Effect of Work Hardening Behaviors on Analysis Results
The effect of work hardening behaviors such as the Bauschinger effect and work hardening rate on the analysis results will be clarified in this section by using different work hardening constitutive relations with different parameters.
To clarify the effect of work hardening behaviors under cyclic loading such as the Bauschinger effect, FE analyses of SPCC with the Hill48 yield criterion with the non-asso-ciated flow rule (non-AFR Hill48, parameters listed in Table 1) and two different hardening rules, which are the isotropic hardening rule and the mixed hardening rule (parameters listed in Table 2), were conducted. It can be concluded from the results (shown in Fig. 6) that the hardening rule has a significant effect on the predicted residual curvature, indi-cating that it is necessary to consider the cyclic hardening behaviors in tension leveling FE analysis. The calculated bending moment during tension leveling with an intermesh of 7.5 mm is shown in Fig. 7. Since the mixed hardening
Fig. 4. Experimental and simulation results with different yield criteria. (Online version in color.)
Fig. 5. Analysis results for SPCC with different values of aniso-tropic parameters with intermesh of 10 mm: (a) H, G and F and (b) N. (Online version in color.)
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rule can describe the cyclic material behaviors such as the Bauschinger effect during cyclic deformation, every time the loading reverses, the yield stress predicted by the mixed hardening rule is lower than that predicted by the isotropic hardening rule (shown in Fig. 8), which leads to a smaller calculated bending moment after several loading reversals. Since the stress–strain relationship described by the mixed hardening rule is closer to reality than that described by the isotropic hardening rule, considering the material cyclic behaviors using the mixed hardening rule can provide higher accuracy for the FE analysis of tension leveling.
Moreover, the mixed hardening rule applied in this investigation can describe the transient hardening behavior after yielding and reyielding observed in the stress–strain curve, in which the work hardening rate changes rapidly. As shown in Fig. 8, after the reyielding due to loading reversal, the work hardening rate in the transient area of the mixed hardening rule is higher than that for the isotropic harden-ing rule. To clarify the effect of the work hardening rate, FE analyses of SPCC with the Hill48 yield criterion with the non-associated flow rule (non-AFR Hill48, parameters listed in Table 1) and the isotropic hardening rule with one, three and ten times the work hardening rate as that for the parameters listed in Table 2, were conducted. When the intermesh is 10 mm, whereas the entry tension in all cases is 40 MPa, the delivery tension values at the exit are 40.303 MPa, 40.579 MPa and 40.045 MPa, respectively, for
the above work hardening rates. It can be seen that when the work hardening rate changes, the delivery tension also changes. Although the flow stress of the material increases with the work hardening rate, the equivalent plastic strain of the material decreases (the work curvature values of the first leveler roll of the three cases are 18.31 m −1, 17.99 m −1 and 16.64 m −1, respectively); thus, the plastic work of the tension leveling process changes, resulting in the change of delivery tension (the difference between the values of the delivery tension and the entry tension can reflect the amount of the plastic work). To obtain FE analysis results of tension leveling with high accuracy, the applied material data in the transient area of the stress–strain curve must be carefully identified to describe the work hardening rate accurately.
4. Tension Leveling Mechanism for Defects of High-strength Steel Strip
Since the effect of tension leveling for a high strength steel strip is considerable, it has been widely used in indus-try to level the defects of high-strength steel strips. Silvestre et al. studied roller leveling for high-strength steel strips by performing both experiments and simulations, and they found that the force required to process high-strength steels is much higher than that required to process mild steel strips.15) During the process design of tension leveling for high-strength steels, although the obtained results for mild steels may not be applicable, FE analysis can provide a large amount of useful information without any experiments.
In this section, the tension leveling mechanism for the curvature defects and waviness defects of an SPFC980 strip (dimensions: length × width × thickness = 1 200 mm × 160 mm × 1 mm) with the multiroll-type tension leveler and the waviness tension leveler is clarified on the basis of FE analysis results to give a deeper insight into the process. For the FE model of tension leveling for high-strength steel strips, the 3D analytical rigid body was used to model the leveler rolls and the second-order brick element was used to model the SPFC980 strip. The Hill48 yield criterion with the non-associated flow rule and the mixed hardening (non AFR Hill48 + mixed hardening, parameters listed in Tables 1 and 2) is used in the FE analyses. The Young’s modulus and Poisson’s ratio of SPFC980 are assumed to be 200 GPa
Fig. 6. Experimental and simulation results with different harden-ing rules. (Online version in color.)
Fig. 7. Calculated bending moment with different hardening rules at measured positions during tension leveling with inter-mesh of 7.5 mm. (Online version in color.)
Fig. 8. Schematic of material behaviors during cyclic loading. (Online version in color.)
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and 0.3, respectively. The applied mesh size for the second-order brick element is 10 mm × 20 mm × 0.25 mm and the applied entry tension is 100 MPa. A schematic of the FE analysis steps for strip defect tension leveling is shown in Figs. 9(a) and 9(b).
4.1. Leveling Mechanism of Multiroll-type Tension Leveler for Tension Leveling of Curvature Defects of SPFC980
There are two main types of curvature defects for metal strips, which are curl (curvature defect in the longitudinal direction) and gutter (curvature defect in the width direc-tion). Considering a strip with curl (or gutter), when it is flattened and placed in the leveler before tension leveling, there are longitudinal direction (or width direction) residual stresses distributed through the thickness of the strip. The residual stress distributions in the thickness direction are assumed to be given by Eqs. (11) and (12) for curl and gutter, respectively. Using residual stresses, tension level-ing FE analyses for curl and gutter were conducted with the arrangement of the multiroll-type tension leveler shown in Fig. 10 set by Hattori et al.16) The intermesh values for each roll from the inlet to the outlet are 4 mm, 3.5 mm, 3 mm, 2.5 mm and 2 mm, respectively. The results after the springback of a 200-mm-long part cut from the strip before and after tension leveling are shown in Figs. 11(a) and 11(b). The results of curl are measured from the width center of the 200-mm-long cutting strip, and the results of the gutter are measured from the half length position of the 200-mm-long cutting strip. The curvature for curl decreases from 1.869 m −1 to 0.061 m −1, and the curvature for gutter decreases from 1.397 m −1 to 0.324 m −1 after the springback after tension leveling. It can be seen from the results that both the curl and gutter defects are leveled by the multiroll-
Fig. 9. Schematic of FE analysis of tension leveling for SPFC980: (a) curvature and (b) waviness defects.
Fig. 10. Multiroll-type tension leveler for tension leveling of curvature defects. (Online version in color.)
Fig. 11. FE analysis results of curvature defect tension leveling for SPFC980: (a) curl (b) gutter. (Online version in color.)
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type tension leveler. The calculated bending moments of the leveled strip during the tension leveling for curl and gutter are shown in Figs. 12(a) and 12(b), respectively. It can be seen that, for both the curl and gutter defects, the initial bending moment caused by the residual stresses becomes almost 0 after tension leveling before the springback, indi-cating that the nonuniform distribution of residual stress in the strip thickness becomes uniform after several cyclic deformations caused by the multiroll-type tension leveler. It can be concluded from the FE analysis results that the multiroll-type tension leveler is suitable for the leveling of
curvature defects of high-strength steel strips owing to its ability to make the nonuniform residual stresses uniform during cyclic deformations.
�� � � � �
xx
yy zz xy yz zx
z z� � � � �� �� � � � �
���
400 0 5 0 5
0
. . .............. (11)
�� � � � �
yy
xx zz xy yz zx
z z� � � � �� �� � � � �
���
300 0 5 0 5
0
. . .............. (12)
4.2. Leveling Mechanism of Waviness Tension Leveler for Tension Leveling of Waviness Defects of SPFC980
The waviness defects of strips are caused by inhomoge-neous elongation during rolling. When the elongation of the central part is larger than that of the edge part, a center buckle defect occurs. In contrast, when the elongation of the edge part is larger than that of the central part, an edge waviness defect occurs. The arrangement of the waviness tension leveler set by Hattori et al. shown in Fig. 13 was applied to conduct the FE analysis of tension leveling for waviness defects of SPFC980.16) The upper surfaces of the strips with central and edge waviness defects are expressed by Eqs. (13) and (14), respectively. The intermesh values for the left and right leveling units are 16 mm and 15.5 mm, respectively. The leveling results for the central and edge waviness defects are shown in Figs. 14(a) and 14(b), respectively. For center buckle leveling, the steepness decreases from 0.02 to 0.0019 after tension leveling. For edge waviness leveling, the steepness decreases from 0.02 to 0.0010 after tension leveling.
In the case of center buckle leveling, the elongations caused by the waviness tension levelers in the central and edge parts of the strip are 0.190% and 0.285%, respectively. For edge waviness leveling, the elongations caused by the waviness tension levelers in the central and edge parts of the strip are 0.288% and 0.185%, respectively. The differ-ence in elongation between the central and edge parts of the strip caused by the waviness tension leveler cancels the elongation difference caused by rolling; thus, flatness is improved. It can be concluded from the FE analysis results for waviness defect tension leveling that the waviness ten-sion leveler is suitable for the leveling of waviness defects Fig. 12. Calculated bending moments during tension leveling for
SPFC980: (a) curl (b) gutter. (Online version in color.)
Fig. 13. Waviness tension leveler for tension leveling of waviness defects. (Online version in color.)
ISIJ International, Vol. 60 (2020), No. 6
© 2020 ISIJ1281
of high-strength steel strips owing to its ability to cancel the difference in elongation between the central and edge parts of the strip caused by rolling.
z x y
yx x y
x
,
. ,
.
� � ��
����
��� � � � � �� �
� �
40
40 500 5 0 1 200 0 40
0 5 0 1 2
sin�
�
000 40 80, � �� �
�
��
�� y
........................................ (13)
z x y
x y
yx x
,
. ,
sin .
� � �� � � �� �
���
��
��� � � �
0 5 0 1 200 0 40
40
40 500 5 0 1 2
�� 000 40 80, � �� �
�
��
��y
........................................ (14)
5. Conclusions
An advanced constitutive model, which can describe the material anisotropy and work hardening behaviors during cyclic loading such as the Bauschinger effect, was con-structed for the FE analysis of tension leveling with high accuracy. To assist the tension leveling process design using FE analysis, case studies were conducted to clarify the rela-
tionship between constitutive relations and analysis results. FE analyses of tension leveling for high-strength steel strips were also conducted to give a deeper insight into the pro-cess. The following conclusions were obtained.
• The FE analysis of tension leveling with a 3D FE model cannot be replaced by that with a 2D plane strain FE model because of the inability of the latter to describe the material flow in the width direction.
• It is necessary to consider the material anisotropy in the FE analysis of tension leveling since it affects the FE analysis results.
• Considering the work hardening behaviors under cyclic loading such as the Bauschinger effect and the tran-sient behavior, applying the mixed hardening rule can pro-vide higher prediction accuracy since the bending moment can be better predicted and the work hardening rate can be described accurately.
• For the tension leveling of curvature defects of high-strength steel strips, a multiroll-type tension leveler is suit-able owing to its ability to make the nonuniform residual stresses uniform during cyclic deformations.
• For the tension leveling of waviness defects of high-strength steel strips, a waviness tension leveler is suitable owing to its ability to cancel the elongation difference in the central and edge parts caused by the previous rolling process.
AcknowledgementsThe authors are sincerely grateful to Professor Hiroshi
Hamasaki of Hiroshima University for providing the experi-mental data of SPFC980. This investigation was conducted as part of a research project supported by the Iron and Steel Institute of Japan.
REFERENCES
1) F. Yoshida and M. Urabe: J. Mater. Process. Technol., 89–90 (1999), 218. https://doi.org/10.1016/S0924-0136(99)00034-5
2) H. Huh, H. Lee, S. Park, G. Kim and S. Nam: J. Mater. Process. Tech-nol., 113 (2001), 714. https://doi.org/10.1016/S0924-0136(01)00661-6
3) J. Morris, S. Hardy and J. Thomas: J. Mater. Process. Technol., 120 (2002), 385. https://doi.org/10.1016/S0924-0136(01)01175-X
4) S. Kim, J. Lee, F. Barlat and M. Lee: J. Mater. Process. Technol., 213 (2013), 1929. https://doi.org/10.1016/j.jmatprotec.2013.05.015
5) S. Sumikawa, A. Ishiwatari, J. Hiramoto and T. Urabe: J. Mater. Process. Technol., 230 (2016), 1. https://doi.org/10.1016/j.jmatprotec.2015.11.004
6) S. Sumikawa, A. Ishiwatari and J. Hiramoto: J. Mater. Process. Tech-nol., 241 (2017), 46. https://doi.org/10.1016/j.jmatprotec.2016.11.005
7) B. Wu, K. Ito, N. Mori, T. Oya, T. Taylor and J. Yanagimoto: Int. J. Precis. Eng. Manuf.-Green Technol., (2019). https://doi.org/10.1007/s40684-019-00032-5
8) B. Wu, H. Wang, T. Taylor and J. Yanagimoto: Int. J. Mech. Sci., 169 (2020), 105320. https://doi.org/10.1016/j.ijmecsci.2019.105320
9) M. Grüber and G. Hirt: Procedia Eng., 207 (2017), 1332. https://doi.org/10.1016/j.proeng.2017.10.892
10) M. Grüber and G. Hirt: Procedia Manuf., 15 (2018), 844. https://doi.org/10.1016/j.promfg.2018.07.180
11) H. Wang, B. Wu and J. Yanagimoto: Steel Res. Int., 90 (2019), 1800401. https://doi.org/10.1002/srin.201800401
12) S. Zang, C. Guo, S. Thuillier and M. Lee: Int. J. Mech. Sci., 53 (2011), 425. https://doi.org/10.1016/j.ijmecsci.2011.03.005
13) J. Yanagimoto: J. Jpn. Soc. Technol. Plast., 55 (2014), 843 (in Japanese). https://doi.org/10.9773/sosei.55.843
14) J. Chakrabarty, W. Lee and K. Chan: Int. J. Mech. Sci., 43 (2001), 1871. https://doi.org/10.1016/S0020-7403(01)00009-1
15) E. Silvestre, E. Sáenz de Argandoña, L. Galdos and J. Mendiguren: Key Eng. Mater., 611–612 (2014), 1753. https://doi.org/10.4028/www.scientific.net/KEM.611-612.1753
16) S. Hattori, Y. Maeda, T. Matsushita, S. Murakami and J. Hata: J. Jpn. Soc. Technol. Plast., 28 (1987), 34 (in Japanese).
Fig. 14. FE analysis results of waviness defect tension leveling for SPFC980: (a) center buckle (b) edge waviness. (Online version in color.)
ISIJ International, Vol. 60 (2020), No. 6
© 2020 ISIJ 1282
Appendix I.
Verification of Bending Moment Continuity when Using Second-order Element in FE Analysis
The type of interpolation function for second-order brick element (shown in Fig. A1) is serendipity interpolation. The interpolation functions Ni(g, h, r) are given as
N g h r
g g h h rr g g h h rr i
i
i i i i i i
, ,
~
� � �
�� � �� � �� � � � �� � �� �1
81 1 1 2 1 8
..................................... (15-a)
N g h r g h h rri i i, , , , ,� � � �� � �� � �� � �� �1
41 1 1 9 11 13 152 i
..................................... (15-b)
N g h r g g h rri i i, , , , ,� � � �� � �� � �� � �� �1
41 1 1 10 12 14 162 i
..................................... (15-c)
N g h r g g h h ri i i, , ~� � � �� � �� � �� � �� �1
41 1 1 17 202 i
..................................... (15-d)
where i is the number of each node. gi, hi and ri are the coor-dinates of node i in the natural coordinate system, which is the normalized form of the local coordinate system (for each axis, the value ranges from −1 to +1).17)
The displacement {u(g, h, r)} of P (g, h, r) in the element 1 (shown in Fig. A1) can be obtained as follows by second-order interpolation using Ni(g, h, r):
u ug h r
u
u
u
N g h r g h r
N g
x
y
z
i ii
i
, , , , , ,� �� � ��
��
��
�
��
��� � � � �� �
�
�� 1
20
,, ,h r
u
u
u
xi
yi
zi
i� �
�
��
��
�
��
��
�� 1
20
... (16)
Since Ni(g, h, r) is the second-order function of each coordinate component, the continuity of displacement in element can be guaranteed. Moreover, as shown in Fig. 15, the adjacent elements share 8 nodes at the element interface, and the displacement at the element interface is controlled by those 8 nodes (when P (g, h, r) locates at the element
interface, the values of interpolation function for the other nodes are all 0 according to Eq. (15)), so the continuity of displacement at element interfaces can also be guaranteed, which is called the C0 continuity.18) For the strain {ε} of P (g, h, r), the result is
��� � �
�
�
����
�
����
�
�
����
�
����
�
��
��
�
������
xx
yy
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xz
x
y
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0 0��
��
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��
��
��
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�
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�
�
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z y
z x
2 2
2 2
2 2
0
0
0
���������
�
��
��
�
��
��
u
u
u
x
y
z
......... (17)
Substituting Eq. (16) into Eq. (17) gives
xx
yy
zz
xy
yz
xz
x
y
u
u
B 6 60
1
1
uu
u
u
u
N
x
N
x
z
x
y
z
1
20
20
20
1 200 0 0 0
0
NNy
N
y
N
z
N
zN
y
N
x
N
y
N
1 20
1 20
1 1 20 20
0 0 0
0 0 0 0
2 20
2 2
xx
N
z
N
y
N
z
N
y
N
z
N
x
N
z
N
0
02 2
02 2
20
2 20
1 1 20 20
1 1 20 20
22
1
1
1
20
20
x
u
u
u
u
u
x
y
z
x
y
uuz20
........................................ (18)
where [B]6×60 is the matrix defining the relationship between the strain and the displacement, whose components con-tain the first derivative of Ni(g, h, r) with respect to each coordinate component. Since the interpolation functions of the second-order element are quadratic, and the adjacent elements share the same nodes at the element interface, the continuity of the first derivative of Ni(g, h, r) in element and at element interfaces can be guaranteed, which is called the C1 continuity.18) Thus, the strain continuity of the second-order brick element at the element interfaces can also be guaranteed.
The stress {σ} in the elastic-plastic FE calculation is calculated in the form of a small increment {dσ} follows:
Fig. A1. Second-order brick element with numbered nodes in nat-ural coordinate system.
ISIJ International, Vol. 60 (2020), No. 6
© 2020 ISIJ1283
d
d
d
d
d
d
d
d
xx
yy
zz
xy
yz
xz
�� ��� � �
�
�
����
�
����
�
�
����
�
����
� � ��
������
D �� � � �
�
�
����
�
����
�
�
����
�
����
D
d
d
d
d
d
d
xx
yy
zz
xy
yz
xz
������
........ (19)
where [D] is the matrix defining the relationship between the stress increment and the strain increment, whose compo-nents contain no derivative of Ni(g, h, r), which means that the continuity of stress at element interfaces is determined by the strain. When calculating the bending moment My at the surface of the element (shown in Fig. A2), the result is given by
M z dAy xxA
� � � ........................... (20)
where A is the area of the surface. Since the bending moment is calculated from the stress, the continuity of the bending moment at the element interfaces is also determined by the strain. Thus, in the FE calculation using the second-order solid element, the continuity of the results of the displacement, strain, stress and bending moment in element and at element interfaces can all be guaranteed.
Appendix II.
Derivation Based on Classical Bending TheoryIn the bending process (shown in Fig. A3), consider-
ing classical bending theory, the distributions of σxx and εxx through the strip thickness at the beginning of plastic deformation are given as
� �xxyn
z
h� � ............................. (21)
� �xx cyz
z
r� � � � .......................... (22)
where n is a constant depending on the yield criterion, h is half of the strip thickness and ρcy is the curvature of the cen-tral layer at the beginning of plastic deformation.19) When plastic deformation occurs, considering Hooke’s law under 3D stress and strain conditions and neglecting the strain in the y and z directions, the distribution of εxx through the
Fig. A2. Schematic of bending moment calculation. Fig. A3. Schematic of bending process.
strip thickness and the stress components at point b at the beginning of plastic deformation are
�
� � �
�xx
yn z
E h��
� �� ��� �
1 2
1
2
.................... (23)
� �xxbyn� ............................ (24-a)
� ����
���yyb zzb xx
yn� �
��
�1 1 ........... (24-b)
where v is Poisson’s ratio, E is Young’s modulus. Substitut-ing Eq. (23) into Eq. (22) gives the curvature of the central layer at the beginning of plastic deformation as
�� � �
�cy
yn
E h�
� �� ��� �
1 2
1
2
...................... (25)
The bending moment at the beginning of plastic deforma-tion is
M zdA zwdzn why
xxA
h
xx
y
h
h� � � �� ��� �
�2
3
2
..... (26)
where w is the width of the strip. It can be seen from Eqs. (25) and (26) that both the curvature of the central layer and the bending moment at the beginning of plastic deformation are related to n. Substituting Eq. (24) into Eq. (1) and solv-ing for n gives
nF G H
G H� �
� �� � �� ��� � �� �
2
3
1
1 2
2
2
�
� ................ (27)
It can be seen that the curvature of the central layer and the bending moment at the beginning of plastic deformation are both determined by the values of anisotropic parameters, which shows the effect of material anisotropy on the bend-ing process.
APPENDICES REFERENCES
17) SIMULIA: ABAQUS 6.14 Documentation, Dassault Systemes, Johnston, RI, (2014).
18) S. Rao: The Finite Element Method in Engineering, Elsevier Butter-worth-Heinemann, Oxford, (2005), 151.
19) S. Timoshenko: History of Strength of Materials, McGraw-Hill, New York, (1953), 25.