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Parametric Study of Edge Crack of Silicon Steel Strip in Cold Rolling based on a Shear Modified GTN Damage Model
Available online at www.sciencedirect.com
ScienceDirect Procedia Materials Science 3 (2014) 1632 – 1637
20th European Conference on Fracture (ECF20)
Parametric study of edge crack of silicon steel strip in cold rolling based on a shear modified GTN damage model Quan Suna, Jianjun Chena*, Xiaoxue Lia, Hongliang Pana a
School of Mechanical & Power Engineering, East China University of Science & Technology, Shanghai 200237, China
Abstract Edge cracking is a commonly observed phenomenon during the cold rolling process of the silicon steel strip which may cause the rupture of strip and reduce the product quality and productivity. In this work, the damage behavior and edge cracking of steel strip in cold rolling is analyzed based on a shear modified Gurson model. Taguchi experimental design method is used to analyze the effects of reduction ratio, roll diameter, tension, friction coefficient on the initiation and propagation of edge cracks. The analysis results reveal that the possibility of edge crack for silicon steel increases with the increasing of reduction ratio, tension, friction coefficient and the decrease of roll diameter. © 2014Published The Authors. Published by Elsevier Ltd.CC BY-NC-ND license. © 2014 by Elsevier Ltd. Open access under Selection andpeer-review peer-review under responsibility ofNorwegian the Norwegian University of Science and Technology Department of Selection and under responsibility of the University of Science and Technology (NTNU), (NTNU), Department of StructuralEngineering. Engineering Structural Keywords: Cold rolling, Edge crack, Damage model, Low stress trixiality, Experimental design
1. Introduction Silicon steel, also called electrical steel, has excellent electrical and magnetic properties. It is widely used as the core material of electromagnetic equipment such as transformer, motor and generator. Cold rolling is one of the essential processes to manufacture silicon steel sheet. However, edge cracks are commonly observed during cold rolling process of silicon steel, which may affect the strip quality and productivity significantly.
* Corresponding author. Tel.: +86-021-64253622; fax: +86-021-64253622. E-mail address: [email protected]
2211-8128 © 2014 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering doi:10.1016/j.mspro.2014.06.263
Quan Sun et al. / Procedia Materials Science 3 (2014) 1632 – 1637
1633
Nomenclature
V eq
Macroscopic von Mises equivalent stress
Vm V ij
Macroscopic hydrostatic stress
sij
Deviatoric stress tensor
V
Flow stress of material matrix Total effective void volume fraction
f* q1 , q2 f f f0 fc
Cauchy stress tensor
Coefficient of the GTN damage model Void volume fraction Void volume fraction Initial void volume fraction Critical void volume
fF fN
Void volume fraction at failure
HN
Mean void nucleation plastic strain
SN
Standard deviation of H N
H
p ij
Macroscopic plastic strain tensor
H
p
Volume fraction of void nucleating particles
Equivalent strain of material matrix w(V ij ) Measurement of current stress state
kw
Material parameter characterize shear damage
J3
The third invariant of the deviatoric stress tensor, J 3
1 3 sij s jk ski
The edge crack is the result of ductile fracture due to plastic flow in cold rolling and the mechanism of its formation is complex. Many researchers have used the finite element method to investigate the influence of rolling process parameters such as reduction ratio, tension, lubrication or friction condition to the initiation and propagation of edge cracks. The key point is to apply a proper material model to describe the ductile damage and fracture behavior in rolling condition. Ghosh et al. (2004) and Zhang et al. (2011) employed the the Cockcroft-Latham model (Cockcroft and Latham, 1968) to predict the edge crack base on the assumption that the failure take place when the maximum tensile principal stress and strain reached a certain value. Mashayekhi et al. (2011) used the Lemaitre damage model (Lemaitre 1992) to characterize the deterioration of material due to damage evolution and final failure during cold rolling, in which the damage is considered to be linear with the equivalent strain and influenced by stress triaxiality. Generally, ductile damage and fracture process of metallic material consists of three stages: micro-voids nucleation, growth and coalescence. The most popular damage model to describe this process is Gurson model (Gurson, 1977), which has been improved by Tvergaard and Needleman (known as GursonTvergaard-Needleman model, GTN). Ghosh et al. (2004) and Yan et al. (2011) used the GTN damage model to predict edge crack in cold rolling. However, Riedel et al. (2007) pointed out that the Gurson model can predict damage evolution and final facture in principle, but it is only suited for high stress triaxiality, while for rolling condition (low stress triaxiality) the accuracy of the prediction is questionable. In this research, a shear modified GTN model proposed by Nahshon and Hutchinson (2008), which account for the effective damage accumulation due to distortion of cavities and inter-cavity interacting under low stress triaxiality, was applied to analyze the ductile damage and fracture behavior in cold rolling, and the influence of rolling process parameters on the edge cracks initiation and propagation are also investigated. The reliability of the
1634
Quan Sun et al. / Procedia Materials Science 3 (2014) 1632 – 1637
simulation results are checked by cold rolling experiments of silicon steel on an experimental non-reversing twohigh mill. 2. The shear modified GTN model The GTN model considered a damage material as a continuum medium with a constitutive equation that includes a damage parameter f * which may vary from zero for undamaged material to one for completely damaged material. The yield surface of GTN damage model can be expressed as follows:
I V ij , V , f
*
2
§ V eq · § 3q2V m · * * 2 ¨ ¸ 2 f q1 cosh ¨ ¸ 1 q1 f V V 2 © ¹ © ¹
0
(1)
The damage variable f * is defined as the function of void volume fraction f
f*
f, ® ¯ f c N ( f f c ),
f d fc f ! fc
(2)
Here, N is the void growth acceleration factor which represents the rapid drop of material load capacity due to void coalescence. And it can be obtained from the void volume fraction at failure f F .
N
fu f c f F f c
(3)
The parameter fu 1 q1 represents the ultimate value of void volume fraction when the material load capacity reduces to zero. The above equations establish the behavior of material when a specific void volume fraction value is considered. During the plastic deformation the change of the void volume fraction is corresponding to the void growth and the nucleation of new voids in the original GTN model. The void growth rate is controlled by the trace of plastic strain increment tensor. fg
(1 f )H kpk
(4)
The nucleation rate of new voids is considered to be governed by normal distribution.
fn
ª 1 § H p H ·2 º N exp « ¨ ¸ »H 2S «¬ 2 © S N ¹ »¼
fN SN
p
(5)
However, in the shear modified GTN model, Nahshon and Hutchinson set up a shear damage mechanism based on phenomenological aspects to capture the material deterioration in shear loading condition. The growth rate of shear damage is defined as: fs
kw w(V ij ) f
sij H ijp
V eq
(6)
Here, k w is a material parameter that defined as the magnitude of the damage growth rate in pure shear state. The shear damage growth rate is influenced by current stress state which is characterized by the third invariant of the deviatoric stress tensor.
1635
Quan Sun et al. / Procedia Materials Science 3 (2014) 1632 – 1637
w(V ij ) 1 27 J 3 2V eq3
2
(7)
Coupled with Nahshon - Hutchinson shear mechanism, the expression for the growth rate of void volume fraction can be written as: f
f g fn fs
(8)
The shear modified GTN damage model is incorporated in the ABAQUS through VUMAT user-defined material subroutine interface. The integration scheme of the constitutive law of the model is detailed by Nahshon and Hutchinson (2009). The material used in this work is a kind of non-oriented silicon steel with 2.1% Silicon in weight approximately. The mechanical and damage parameters of this material has been determined by tensile experimental data and SEM observations by Yan et al. (2011), and the shear damage parameter k w is obtained from shear test of a special shear specimen by the authors (2013). 3. Finite element analysis 3.1. Finite element model Three-dimension FE analyses were carried out to predict the onset of crack in cold rolling process by using the shear modified GTN model. The schematic diagram of the cold rolling process is illustrated in Fig. 1, where X is the rolling direction, Y is the strip thickness direction and Z is the transverse direction. Owing to the symmetry, only a quarter of the whole model was modeled. The dimension of the strip was selected as L=30mm, W=40mm and H=2.6mm. A notch with width 0.2mm and length 3mm was preset in the strip edge in the FE model. The strip was meshed with Eight-node brick element with reduced integration (C3D8R) and the work roll was modeled as analytical rigid body. A local mesh refinement was adopted for the region around the edge notch tip (Fig. 1). An initial velocity slightly less than the roll tangential velocity (1m/s) was assigned to the roll strip to make sure that the strip can enter into the roll gap smoothly. Between the roll and strip, a surface-to-surface contact with Coulomb friction condition was defined in the cold rolling simulation. The rolling tension was set as pressure at the front and rear ends of the strip.
Fig. 1. A schema of cold rolling process and mesh refinement around the notch tip.
4. Results and discussion 4.1. The initiation and propagation of edge cracks The plastic flow of material in rolling gap leads to the material deterioration and finally failure. In numerical simulation, the damage increase with the plastic strain. When the damage reached the critical value, the element will
1636
Quan Sun et al. / Procedia Materials Science 3 (2014) 1632 – 1637
removed and the crack appears. Fig.2 shows the comparison of simulation and experiment result of the notch cracking morphology after rolling. Both the simulation and experiment show that two cracks appear around the notch tip and extend along the direction with an angle about 45° and 135° to the rolling direction, which indicates that the shear modified GTN damage model is efficient to predict damage and fracture behavior of steel strip in rolling condition. In this work, the crack length was measured as the maximum distance between the removed elements.
Fig. 2. Notch cracking behavior of (a) simulation and (b) experiment after rolling ( A=40, B=0, C=180, D=0.1)
4.2. Taguchi experimental design analysis Taguchi experimental design (Orthogonal experimental design) is an effective method to analyze the effects of multiple variables. In present work, four processing parameters such as reduction ratio, tension, work roll diameter and friction coefficient which were considered to be factors having great effect on edge cracks in the industrial manufacture, were taken into account. Therefore, a 4-factor-4-level Taguchi design was carried out in which A represents for work roll diameter with various levels of 180, 240, 300, 360 mm; B for reduction ratio of 25%, 30%, 35%, 40% ; C for friction coefficient of 0.15, 0.2, 0.25, 0.3 and D for tension of 0, 50, 100, 150 MPa. In the Taguchi experimental design analyses, 16 groups test experimental with different combinations of rolling process parameters were conducted. The scheme of the experimental design analysis was determined by the statistical software Minitab. The crack length was set to be the response. The responses were obtained via FE method based on the shear modified GTN damage model. The experimental cases and each result are listed in table 1. The main effects plot of each parameter is shown in Fig.3. Referring to Fig.3, it can be found that the factors affect the response by different degrees and the edge crack length increases with the increasing of reduction ratio, tension and friction coefficient and the decrease of roll diameter. The rolling reduction ratio and tension are the most two important factors that influence the damage and fracture of steel strip in cold rolling. If a relatively high value of reduction ratio and tension are necessary to keep normal production in industry, employing a bigger work roll is also a good way to prevent the occurrence of edge crack. Table 1. Taguchi experiment scheme and response results
Case A (mm) B (%) C D (MPa) Response (10-1mm)
1 180 25 0.15 0
2 180 30 0.20 50
3 180 35 0.25 100
4 180 40 0.30 150
5 240 25 0.20 100
6 240 30 0.15 150
7 240 35 0.30 0
8 240 40 0.25 50
9 300 25 0.25 150
10 300 30 0.30 100
11 300 35 0.15 50
12 300 40 0.20 0
13 360 25 0.30 50
14 360 30 0.25 0
15 360 35 0.20 150
16 360 40 0.15 100
0.65
1.80
3.40
6.40
2.35
3.85
2.00
3.20
3.45
2.55
2.25
2.45
1.05
0.55
4.30
3.55
Quan Sun et al. / Procedia Materials Science 3 (2014) 1632 – 1637
1637
Fig. 3. The plot of main effects for means in the 4-factor-4-level Taguchi experimental design.
5. Conclusion The effects of reduction ratio, tension, work roll diameter and friction coefficient on the onset of edge crack have been investigated by using FEM method based on the shear modified GTN damage model combine with Taguchi experimental design analysis. The results shows that steel strip with edge notch is easy to cause edge crack after rolling. The crack length increases with an increasing of reduction ratio, tension and friction coefficient, and a bigger work roll is beneficial to reduce the edge cracking. Acknowledgements This work was financially supported by the Natural Science Foundation of China (51105143, 51375164), the Fundamental Research Funds for the Central Universities (1114036) and Shanghai University Young Teachers Training Funds (YG0142129). References Cockcroft, M., Latham, D., 1968. Ductility and the workability of metals. Journal of the Institute of Metals 96(1), 33–39. Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth, Part I. Yield criteria and flow rules for porous ductile media. Journal of Engineering Materials and Technology 99, 2–15. Ghosh, S., Li, M., Gardiner, D., 2004. A computational and experimental study of cold rolling of aluminum alloys with edge cracking. Journal of Manufacturing Science and Engineering-Transactions of the Asme 126(1),74–82. Lemaitre, J., 1992. A course on damage mechanics. Germany: Spinger-Verlag. Mashayekhi, M., Torabian, N., Poursina, M., 2011. Continuum damage mechanics analysis of strip tearing in a tandem cold rolling process. Simulation Modelling Practice and Theory 19(2), 612–625. Nahshon, K., Hutchinson, J.W., 2008. Modification of the Gurson Model for shear failure. European Journal of Mechanics a-Solids 27(1), 1–17. Riedel, H., Andrieux, F., Walde, T., Karhausen, K.F., 2007. The formation of edge cracks during rolling of metal sheet. Steel Research International 78(10-11), 818–824. Sun, Q., Yan, Y.X., Chen, J.J., Li, X.X., Pan, H.L., 2013. Implementation of a shear modified GTN damage model and its application in cold rolling. Advanced Materials research 815, 758–764 Yan, Y.X., Sun, Q., Chen, J.J., Pan, H.L., 2013. The initiation and propagation of edge cracks of silicon steel during tandem cold rolling process based on the Gurson–Tvergaard–Needleman damage model. Journal of Materials Processing Technology 213(4), 598–605. Zhang, D.F., Dai, Q.W., Fang, L., Xu, X.X., 2011. Prediction of edge cracks and plastic-damage analysis of Mg alloy sheet in rolling. Transactions of Nonferrous Metals Society of China 21(5), 1112–1117.
ScienceDirect Procedia Materials Science 3 (2014) 1632 – 1637
20th European Conference on Fracture (ECF20)
Parametric study of edge crack of silicon steel strip in cold rolling based on a shear modified GTN damage model Quan Suna, Jianjun Chena*, Xiaoxue Lia, Hongliang Pana a
School of Mechanical & Power Engineering, East China University of Science & Technology, Shanghai 200237, China
Abstract Edge cracking is a commonly observed phenomenon during the cold rolling process of the silicon steel strip which may cause the rupture of strip and reduce the product quality and productivity. In this work, the damage behavior and edge cracking of steel strip in cold rolling is analyzed based on a shear modified Gurson model. Taguchi experimental design method is used to analyze the effects of reduction ratio, roll diameter, tension, friction coefficient on the initiation and propagation of edge cracks. The analysis results reveal that the possibility of edge crack for silicon steel increases with the increasing of reduction ratio, tension, friction coefficient and the decrease of roll diameter. © 2014Published The Authors. Published by Elsevier Ltd.CC BY-NC-ND license. © 2014 by Elsevier Ltd. Open access under Selection andpeer-review peer-review under responsibility ofNorwegian the Norwegian University of Science and Technology Department of Selection and under responsibility of the University of Science and Technology (NTNU), (NTNU), Department of StructuralEngineering. Engineering Structural Keywords: Cold rolling, Edge crack, Damage model, Low stress trixiality, Experimental design
1. Introduction Silicon steel, also called electrical steel, has excellent electrical and magnetic properties. It is widely used as the core material of electromagnetic equipment such as transformer, motor and generator. Cold rolling is one of the essential processes to manufacture silicon steel sheet. However, edge cracks are commonly observed during cold rolling process of silicon steel, which may affect the strip quality and productivity significantly.
* Corresponding author. Tel.: +86-021-64253622; fax: +86-021-64253622. E-mail address: [email protected]
2211-8128 © 2014 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering doi:10.1016/j.mspro.2014.06.263
Quan Sun et al. / Procedia Materials Science 3 (2014) 1632 – 1637
1633
Nomenclature
V eq
Macroscopic von Mises equivalent stress
Vm V ij
Macroscopic hydrostatic stress
sij
Deviatoric stress tensor
V
Flow stress of material matrix Total effective void volume fraction
f* q1 , q2 f f f0 fc
Cauchy stress tensor
Coefficient of the GTN damage model Void volume fraction Void volume fraction Initial void volume fraction Critical void volume
fF fN
Void volume fraction at failure
HN
Mean void nucleation plastic strain
SN
Standard deviation of H N
H
p ij
Macroscopic plastic strain tensor
H
p
Volume fraction of void nucleating particles
Equivalent strain of material matrix w(V ij ) Measurement of current stress state
kw
Material parameter characterize shear damage
J3
The third invariant of the deviatoric stress tensor, J 3
1 3 sij s jk ski
The edge crack is the result of ductile fracture due to plastic flow in cold rolling and the mechanism of its formation is complex. Many researchers have used the finite element method to investigate the influence of rolling process parameters such as reduction ratio, tension, lubrication or friction condition to the initiation and propagation of edge cracks. The key point is to apply a proper material model to describe the ductile damage and fracture behavior in rolling condition. Ghosh et al. (2004) and Zhang et al. (2011) employed the the Cockcroft-Latham model (Cockcroft and Latham, 1968) to predict the edge crack base on the assumption that the failure take place when the maximum tensile principal stress and strain reached a certain value. Mashayekhi et al. (2011) used the Lemaitre damage model (Lemaitre 1992) to characterize the deterioration of material due to damage evolution and final failure during cold rolling, in which the damage is considered to be linear with the equivalent strain and influenced by stress triaxiality. Generally, ductile damage and fracture process of metallic material consists of three stages: micro-voids nucleation, growth and coalescence. The most popular damage model to describe this process is Gurson model (Gurson, 1977), which has been improved by Tvergaard and Needleman (known as GursonTvergaard-Needleman model, GTN). Ghosh et al. (2004) and Yan et al. (2011) used the GTN damage model to predict edge crack in cold rolling. However, Riedel et al. (2007) pointed out that the Gurson model can predict damage evolution and final facture in principle, but it is only suited for high stress triaxiality, while for rolling condition (low stress triaxiality) the accuracy of the prediction is questionable. In this research, a shear modified GTN model proposed by Nahshon and Hutchinson (2008), which account for the effective damage accumulation due to distortion of cavities and inter-cavity interacting under low stress triaxiality, was applied to analyze the ductile damage and fracture behavior in cold rolling, and the influence of rolling process parameters on the edge cracks initiation and propagation are also investigated. The reliability of the
1634
Quan Sun et al. / Procedia Materials Science 3 (2014) 1632 – 1637
simulation results are checked by cold rolling experiments of silicon steel on an experimental non-reversing twohigh mill. 2. The shear modified GTN model The GTN model considered a damage material as a continuum medium with a constitutive equation that includes a damage parameter f * which may vary from zero for undamaged material to one for completely damaged material. The yield surface of GTN damage model can be expressed as follows:
I V ij , V , f
*
2
§ V eq · § 3q2V m · * * 2 ¨ ¸ 2 f q1 cosh ¨ ¸ 1 q1 f V V 2 © ¹ © ¹
0
(1)
The damage variable f * is defined as the function of void volume fraction f
f*
f, ® ¯ f c N ( f f c ),
f d fc f ! fc
(2)
Here, N is the void growth acceleration factor which represents the rapid drop of material load capacity due to void coalescence. And it can be obtained from the void volume fraction at failure f F .
N
fu f c f F f c
(3)
The parameter fu 1 q1 represents the ultimate value of void volume fraction when the material load capacity reduces to zero. The above equations establish the behavior of material when a specific void volume fraction value is considered. During the plastic deformation the change of the void volume fraction is corresponding to the void growth and the nucleation of new voids in the original GTN model. The void growth rate is controlled by the trace of plastic strain increment tensor. fg
(1 f )H kpk
(4)
The nucleation rate of new voids is considered to be governed by normal distribution.
fn
ª 1 § H p H ·2 º N exp « ¨ ¸ »H 2S «¬ 2 © S N ¹ »¼
fN SN
p
(5)
However, in the shear modified GTN model, Nahshon and Hutchinson set up a shear damage mechanism based on phenomenological aspects to capture the material deterioration in shear loading condition. The growth rate of shear damage is defined as: fs
kw w(V ij ) f
sij H ijp
V eq
(6)
Here, k w is a material parameter that defined as the magnitude of the damage growth rate in pure shear state. The shear damage growth rate is influenced by current stress state which is characterized by the third invariant of the deviatoric stress tensor.
1635
Quan Sun et al. / Procedia Materials Science 3 (2014) 1632 – 1637
w(V ij ) 1 27 J 3 2V eq3
2
(7)
Coupled with Nahshon - Hutchinson shear mechanism, the expression for the growth rate of void volume fraction can be written as: f
f g fn fs
(8)
The shear modified GTN damage model is incorporated in the ABAQUS through VUMAT user-defined material subroutine interface. The integration scheme of the constitutive law of the model is detailed by Nahshon and Hutchinson (2009). The material used in this work is a kind of non-oriented silicon steel with 2.1% Silicon in weight approximately. The mechanical and damage parameters of this material has been determined by tensile experimental data and SEM observations by Yan et al. (2011), and the shear damage parameter k w is obtained from shear test of a special shear specimen by the authors (2013). 3. Finite element analysis 3.1. Finite element model Three-dimension FE analyses were carried out to predict the onset of crack in cold rolling process by using the shear modified GTN model. The schematic diagram of the cold rolling process is illustrated in Fig. 1, where X is the rolling direction, Y is the strip thickness direction and Z is the transverse direction. Owing to the symmetry, only a quarter of the whole model was modeled. The dimension of the strip was selected as L=30mm, W=40mm and H=2.6mm. A notch with width 0.2mm and length 3mm was preset in the strip edge in the FE model. The strip was meshed with Eight-node brick element with reduced integration (C3D8R) and the work roll was modeled as analytical rigid body. A local mesh refinement was adopted for the region around the edge notch tip (Fig. 1). An initial velocity slightly less than the roll tangential velocity (1m/s) was assigned to the roll strip to make sure that the strip can enter into the roll gap smoothly. Between the roll and strip, a surface-to-surface contact with Coulomb friction condition was defined in the cold rolling simulation. The rolling tension was set as pressure at the front and rear ends of the strip.
Fig. 1. A schema of cold rolling process and mesh refinement around the notch tip.
4. Results and discussion 4.1. The initiation and propagation of edge cracks The plastic flow of material in rolling gap leads to the material deterioration and finally failure. In numerical simulation, the damage increase with the plastic strain. When the damage reached the critical value, the element will
1636
Quan Sun et al. / Procedia Materials Science 3 (2014) 1632 – 1637
removed and the crack appears. Fig.2 shows the comparison of simulation and experiment result of the notch cracking morphology after rolling. Both the simulation and experiment show that two cracks appear around the notch tip and extend along the direction with an angle about 45° and 135° to the rolling direction, which indicates that the shear modified GTN damage model is efficient to predict damage and fracture behavior of steel strip in rolling condition. In this work, the crack length was measured as the maximum distance between the removed elements.
Fig. 2. Notch cracking behavior of (a) simulation and (b) experiment after rolling ( A=40, B=0, C=180, D=0.1)
4.2. Taguchi experimental design analysis Taguchi experimental design (Orthogonal experimental design) is an effective method to analyze the effects of multiple variables. In present work, four processing parameters such as reduction ratio, tension, work roll diameter and friction coefficient which were considered to be factors having great effect on edge cracks in the industrial manufacture, were taken into account. Therefore, a 4-factor-4-level Taguchi design was carried out in which A represents for work roll diameter with various levels of 180, 240, 300, 360 mm; B for reduction ratio of 25%, 30%, 35%, 40% ; C for friction coefficient of 0.15, 0.2, 0.25, 0.3 and D for tension of 0, 50, 100, 150 MPa. In the Taguchi experimental design analyses, 16 groups test experimental with different combinations of rolling process parameters were conducted. The scheme of the experimental design analysis was determined by the statistical software Minitab. The crack length was set to be the response. The responses were obtained via FE method based on the shear modified GTN damage model. The experimental cases and each result are listed in table 1. The main effects plot of each parameter is shown in Fig.3. Referring to Fig.3, it can be found that the factors affect the response by different degrees and the edge crack length increases with the increasing of reduction ratio, tension and friction coefficient and the decrease of roll diameter. The rolling reduction ratio and tension are the most two important factors that influence the damage and fracture of steel strip in cold rolling. If a relatively high value of reduction ratio and tension are necessary to keep normal production in industry, employing a bigger work roll is also a good way to prevent the occurrence of edge crack. Table 1. Taguchi experiment scheme and response results
Case A (mm) B (%) C D (MPa) Response (10-1mm)
1 180 25 0.15 0
2 180 30 0.20 50
3 180 35 0.25 100
4 180 40 0.30 150
5 240 25 0.20 100
6 240 30 0.15 150
7 240 35 0.30 0
8 240 40 0.25 50
9 300 25 0.25 150
10 300 30 0.30 100
11 300 35 0.15 50
12 300 40 0.20 0
13 360 25 0.30 50
14 360 30 0.25 0
15 360 35 0.20 150
16 360 40 0.15 100
0.65
1.80
3.40
6.40
2.35
3.85
2.00
3.20
3.45
2.55
2.25
2.45
1.05
0.55
4.30
3.55
Quan Sun et al. / Procedia Materials Science 3 (2014) 1632 – 1637
1637
Fig. 3. The plot of main effects for means in the 4-factor-4-level Taguchi experimental design.
5. Conclusion The effects of reduction ratio, tension, work roll diameter and friction coefficient on the onset of edge crack have been investigated by using FEM method based on the shear modified GTN damage model combine with Taguchi experimental design analysis. The results shows that steel strip with edge notch is easy to cause edge crack after rolling. The crack length increases with an increasing of reduction ratio, tension and friction coefficient, and a bigger work roll is beneficial to reduce the edge cracking. Acknowledgements This work was financially supported by the Natural Science Foundation of China (51105143, 51375164), the Fundamental Research Funds for the Central Universities (1114036) and Shanghai University Young Teachers Training Funds (YG0142129). References Cockcroft, M., Latham, D., 1968. Ductility and the workability of metals. Journal of the Institute of Metals 96(1), 33–39. Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth, Part I. Yield criteria and flow rules for porous ductile media. Journal of Engineering Materials and Technology 99, 2–15. Ghosh, S., Li, M., Gardiner, D., 2004. A computational and experimental study of cold rolling of aluminum alloys with edge cracking. Journal of Manufacturing Science and Engineering-Transactions of the Asme 126(1),74–82. Lemaitre, J., 1992. A course on damage mechanics. Germany: Spinger-Verlag. Mashayekhi, M., Torabian, N., Poursina, M., 2011. Continuum damage mechanics analysis of strip tearing in a tandem cold rolling process. Simulation Modelling Practice and Theory 19(2), 612–625. Nahshon, K., Hutchinson, J.W., 2008. Modification of the Gurson Model for shear failure. European Journal of Mechanics a-Solids 27(1), 1–17. Riedel, H., Andrieux, F., Walde, T., Karhausen, K.F., 2007. The formation of edge cracks during rolling of metal sheet. Steel Research International 78(10-11), 818–824. Sun, Q., Yan, Y.X., Chen, J.J., Li, X.X., Pan, H.L., 2013. Implementation of a shear modified GTN damage model and its application in cold rolling. Advanced Materials research 815, 758–764 Yan, Y.X., Sun, Q., Chen, J.J., Pan, H.L., 2013. The initiation and propagation of edge cracks of silicon steel during tandem cold rolling process based on the Gurson–Tvergaard–Needleman damage model. Journal of Materials Processing Technology 213(4), 598–605. Zhang, D.F., Dai, Q.W., Fang, L., Xu, X.X., 2011. Prediction of edge cracks and plastic-damage analysis of Mg alloy sheet in rolling. Transactions of Nonferrous Metals Society of China 21(5), 1112–1117.