Continuum damage mechanics analysis of strip tearing in a tandem cold rolling process

Simulation Modelling Practice and Theory 19 (2011) 612–625

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Continuum damage mechanics analysis of strip tearing in a tandem cold rolling process M. Mashayekhi a,⇑, N. Torabian b, M. Poursina c a

Mechanical Engineering Department, Isfahan University of Technology, Iran Faculty of Engineering, Shahrekord University, Shahrekord, Iran c Faculty of Engineering, University of Isfahan, Isfahan, Iran b

a r t i c l e

i n f o

Article history: Received 7 August 2010 Received in revised form 5 October 2010 Accepted 7 October 2010 Available online 13 October 2010 Keywords: Tandem cold rolling Strip tearing Continuum damage mechanics

a b s t r a c t Strip tearing during tandem cold rolling is one of the manufacturing issues in the rolling industry that can significantly increase production costs. In this paper, an explicit finite element code coupled with the improved Lemaitre damage model is developed to predict strip tearing in a five-stand tandem rolling mill. The simulation results are validated using experimental data from an industrial rolling mill, and there is good agreement between the numerical and experimental results. Some factors related to strip tearing, such as friction coefficient variations, thickness difference between two welded strips and reduction schedule, are introduced, and their influence on the damage evolution through the strip during the process is investigated by using damage-coupled finite element simulations. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The rolling process has an important role in the manufacture of a wide variety of products because of its high accuracy, efficiency and production rate. Sheets and strips can be rolled either in single stand or tandem mills. However, because of an increasing demand for rolled products, the focus has shifted towards tandem rolling mills because of their high speed as well as the high quality and accuracy of their products. Similar to other metal forming processes, the final product of cold rolling can exhibit some mechanical defects. Johnson and Mamalis [1] tabulated various defects that were observed in industrial metal forming processes, including plane strain rolling. Based on industrial reports, common defects in the sheet metal rolling process are alligatoring, edge cracking, central burst, surface defects and buckling of the strip. Additionally, tandem rolling involves another type of problem: strip tearing at the rolling stands. Among these defects, strip tearing requires special consideration, because it not only significantly increases the production costs but can also cause serious damage to the rolls and mill accessories. Generally speaking, product defects in metal forming processes can be addressed by employing two different categories of approaches. The first category, which is referred to as the traditional approach, consists of stress-and-strain based methods and traditional fracture mechanics. Although this type of approach has been widely used in the literature, it had some limitations in predicting fracture initiation [2–5]. Hubert et al. [6] performed experiments to investigate the strip edge cracking phenomenon in steel rolling on an experimental rolling stand by using the Upsetting Rolling Test (URT). Scan Electron Microscope (SEM) micrographs and optical surface profiler topographies indicate that forward strip has a large effect on the opening of cracks. Furthermore, the edge strain hardening induced by the edge-trimming process is a major cause of edge cracking. Additionally, Hubert et al. ⇑ Corresponding author. Tel.: +98 311 3915216; fax: +98 311 3912628. E-mail address: [email protected] (M. Mashayekhi). 1569-190X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2010.10.003

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developed a numerical simulation in which the deviatoric stress state, which is described using the third stress tensor invariant, is considered as the main factor that affects the specimen edge. The numerical results predict a significant stress gradient along the strip edges (where cracks are found in practice). The second type of approach, continuum damage mechanics (CDM), was developed within the last two decades and is a powerful tool for predicting material failure. The phenomenon including nucleation, growth and coalescence of microvoids and microcracks induced by large deformations in metals is called ‘‘ductile damage”, and CDM is the study of this deterioration process by using mechanical variables [7]. However, the literature review revealed that there have only been a few attempts to employ CDM to predict mechanical defects in the final products of sheet metal rolling process [8]. Ghosh et al. [9] utilized the Gurson and the Cockroft–Latham damage models incorporated in a three-dimensional finite element (FE) simulation to predict edge cracking, which is a commonly observed phenomenon, in the single-stand cold rolling of aluminum alloys. The damage evolution predicted by both damage models was consistent and exhibited considerable damage localization near the free edge and the central symmetry plane (where edge cracking is typically found in industrial cold rolling). Rajak and Reddy [8] developed a ductile fracture criterion based on damage evolution along with a simple criterion based on the nature and magnitude of hydrostatic stress to predict the initiation of central burst and split-end in plane strain, single-stand rolling. Additionally, only a limited amount of research has addressed strip tearing during tandem rolling. Muscat-Fenech and Atkins [10] studied the influence of anisotropy in the tearing of cold-rolled sheets. Pimenov et al. [11] developed an automatic system in order to reduce strip tearing during the stopping and starting of a tandem cold rolling mill. However, the application of continuum damage mechanics for the prediction of strip tearing in tandem cold rolling is absent from the literature. In this study, the strip tearing phenomenon in an industrial cold rolling mill is investigated by employing continuum damage mechanics. A five-stand tandem cold rolling mill at Isfahan Mobarakeh Steel Company (IMSC) was simulated using an explicit finite element code. An improved version of the Lemaitre damage model was incorporated in the FE simulations to predict damage distribution through the strip during the rolling process. Afterwards, some factors related to strip tearing were introduced, and their influence on damage evolution was examined by using numerical simulations. The simulations were performed and validated using experimental data from the industrial rolling mill. Section 2 introduces the properties of the IMSC industrial rolling mill. The fundamentals of damage mechanics are presented in Section 3. Section 4 addresses the material parameters determination process. Additionally, numerical simulations of the rolling mill are provided in Section 5. Finally, in Section 6, the effective factors in the strip tearing phenomenon are studied. 2. Industrial tandem cold rolling mill A schematic diagram of a five-stand tandem cold rolling mill at IMSC is shown in Fig. 1; each rolling stand consists of a pair of work rolls beside a pair of backup rolls. The upper rolls are vertically adjustable, which allows the roll gap to be set to any value. The distance between two adjacent stands is 4 m. The maximum value of the strip linear velocity at each stand is also shown in this figure. Table 1 shows the mechanical properties of the rolling mill. Additionally, the number of motors that power the rolls in addition to their nominal power and the angular velocity of the work rolls are listed for each rolling stand in Table 2.

Vmax 854 m/min

Vmax 958 m/min

Vmax 1112 m/min

Vmax 1292 m/min

Vmax 810 m/min

Vmax 1812 m/min

Work roll

Backup roll

Fig. 1. Schematic diagram of an industrial five-stand tandem cold rolling mill.

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M. Mashayekhi et al. / Simulation Modelling Practice and Theory 19 (2011) 612–625 Table 1 Mechanical properties of the industrial rolling mill [12]. Work rolls diameter (mm) Maximum roll force (ton) Initial strip thickness (mm) Final strip thickness (mm)

510–585 3000 1.5–3.5 0.18–3

Table 2 Nominal power and angular velocity of each stand [12]. Stand no.

1

2

3

4

5

Power (kW) Angular velocity (rpm)

2  1950 260–715

2  2400 255–650

2  2400 255–650

2  2400 255–650

3  1950 260–715

Fig. 2. Industrial strip tearing sample [12].

Based on industrial reports, several types of defects were observed in the final products of the rolling mill such as edge cracking, surface defects and alligatoring. However, strip tearing, which mostly occurs at Stands 4 and 5, accounts for a large portion of the amount of productivity that is lost. Therefore, it is necessary to predict strip tearing and analyze the factors that influence this phenomenon. Fig. 2 shows a sample strip tearing case from the industrial mill. In the following sections, the strip tearing phenomenon during tandem cold rolling is studied by using numerical simulations. 3. Continuum damage mechanics model for ductile fracture prediction During the rolling process, the workpiece is subjected to both strain hardening and ductile damage. These two competing mechanisms determine the feasibility of the process and the defects of the rolled strip. In the rolling process, crack closure effects have a strong influence on damage evolution. Therefore, it is necessary to split the tensile and compression stresses in damage evolution and address them separately. In this paper, the improved Lemaitre damage model proposed by Lemaitre [7] is implemented and utilized. The improved model takes into account a different damage growth in tension and compression and considers the crack closure effects by introducing a crack closure parameter h [7,13], while in the standard Lemaitre damage model the tension and compression responses are not differentiated in damage growth [14,15]. The damage variable, D, is defined as the net area of a unit surface that is cut by a given plane and corrected for the presence of existing cracks and cavities. This parameter is bounded by 0 (undamaged state) and 1 (rupture) [16]. In fact, for ductile materials, rupture corresponds to a critical value of damage, Dcr, ranging from 0.2 to 0.5, depending on the material and the loading conditions [7]. By assuming a homogeneous distribution of microvoids and the hypothesis of strain equivalence, which states that the strain behavior of a damaged material can be represented as constitutive equations of the virgin mate~ , can rial in the potential of which the stress is simply replaced by the effective stress, the effective stress tensor in tension, r be represented as:

r~ ¼

r 1D

;

ð1Þ

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~ can be represented as: where r is the stress tensor for the undamaged material. Under compressive stresses, r

r~ ¼

r 1  hD

ð2Þ

;

where h represents the crack closure parameter. The parameter h can take different values in the range of 0 6 h 6 1 depending on the material and the loading conditions [7,17]. In this study, h was determined in a similar way as Ref. [14]; different values were attributed to the h parameter and it was observed that for h = 0.38 the model predicted correctly the fracture location comparing to the experimental observations [12]. Therefore, the h parameter was set to 0.38 in all simulations. The evolution equation for internal variables can be derived by assuming the existence of a potential of dissipation, W, which represents a scalar convex function of the state variables as:

W¼Uþ

 sþ1 r Y : ð1  DÞðs þ 1Þ r

ð3Þ

For a process that accounts for isotropic hardening and isotropic damage, where r and s are material- and temperaturedependent properties, U and Y are, respectively, the yield function and the damage strain energy release rate, given by:





U r; epeq ; D ¼

req ð1  DÞ



h



i

r0Y þ R epeq ;

ð4Þ

and

Y ¼

1 2Eð1  DÞ2

½ð1 þ v Þrþ : rþ  v htr ri2  þ

h 2Eð1  hDÞ2

½ð1 þ v Þr : r  v htr ri2 ;

ð5Þ

p 0 whereprffiffiffiffiffiffiffiffi Y is the initial yield stress, R represents the radial growth of the yield surface, pffiffiffiffiffiffiffiffi eeq is the equivalent plastic strain, p p p eeq ¼ 2=3ke k; e is the plastic strain tensor, req is the equivalent stress, req ¼ 3=2ksk; s is the deviatoric stress tensor, r+ and r are the tensile and compressive components of r, respectively, E is the Young’s modulus and m is Poisson’s ratio. By the hypothesis of generalized normality, the plastic flow equation is:

e_ p ¼ c_

3 s ; 2 req

ð6Þ

and the evolution laws of the internal variables are:

@W ¼ c_ ; @R  s @W 1 Y ; D_ ¼ c_ ¼ c_ 1D r @Y

e_ peq ¼ c_

ð7Þ

where c_ is the plastic consistency parameter, which is subject to the Kuhn–Tucker conditions for loading and unloading:

c_ P 0; U 6 0;

c_ U ¼ 0:

ð8Þ

Full details of the algorithm for the numerical integration can be found in Mashayekhi et al. [18,19]. 4. The material model The strip is composed of St14 steel (DIN 1623) and the material composition is shown in Table 3. Standard tensile and Vickers micro-hardness tests were performed to determine the mechanical properties and damage parameters of the St14 steel. 4.1. Identification of the material properties A standard tensile specimen was selected according to ASTM Standard E8-04 [20]. The geometry of the specimen is shown in Fig. 3. The specimen thickness was 0.8 mm and the rate of displacement was adjusted to 1 mm/min. Fig. 4 shows the experimental results that were obtained from the tensile test, which were regressed by a power hardening law that is defined as:

rY ¼ K enp :

ð9Þ

Table 3 Material composition of alloying elements of St14 steel. Alloying element

C

Si

Mn

P

S

Cu

Al

N

Nb

Mo

Ti

V

%Weight

0.39

0.16

2.32

0.11

0.07

0.36

0.50

0.25

0.01

0.01

0.02

0.04

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Fig. 3. Standard tensile test specimen (dimensions are in mm).

300

250

σY = K ε pn

Stress (MPa)

200

150

100 Exp. data Curve fit

50

0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Strain Fig. 4. Stress–strain curve for the St14 steel.

Table 4 Mechanical and damage parameters for the St14 steel. Young’s modulus, E (GPa) Initial yield stress, ry0 (MPa) Ultimate stress, ru (MPa) Elongation, eu (%) Hardening coefficient, K (MPa) Hardening power, n Damage parameter, r (MPa) Damage parameter, s

180 159 283 44 630 0.36 2.532 1

The mechanical properties of the material are shown in Table 4. 4.2. Identification of the damage parameters A simple method to quantify the isotropic elastic damage variable is to measure the decrease in the micro-hardness of the material. Successive loading of the material allows for the measurement of the different stages of damage, which is then calculated using the following relation [7]:

D¼1

HD ; H0

ð10Þ

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0.5

0.45

Damage

0.4

0.35

0.3

0.25

Exp. data

0.2

Damage evolution law

0.15 0

0.05

0.1

0.15

0.2 0.25 Plastic strain

0.3

0.35

0.4

0.45

Fig. 5. Damage law for the material.

where HD is the micro-hardness of the damaged material, which is determined from measurements, and H0 is the microhardness of the virgin material. In this study, micro-hardness tests were performed using a Vickers indenter under a 0.2 N indentation load. Using Eq. (10) and the magnitude of micro-hardness, the magnitude of damage parameters can be determined. The values of damage that were calculated versus the plastic strain are shown in Fig. 5. From this figure, the damage parameters of the material can be determined because, according to the experimental results in Lemaitre [7,16], the parameter s is assumed to be 1 for this material, and the parameter r can be calculated from the following relation:

r¼

r2 2Eð1  DÞ2 ddD ep

ð11Þ

:

Using the slope of the regression line, the value of dD/dep in Eq. (11) was determined, and the mean value of r was found to be 2.532 MPa. Additionally, from Fig. 5, the magnitude of the critical damage parameter is Dcr = 0.434. 5. Numerical simulation of the five-stand tandem rolling mill 5.1. Finite element model To study the five-stand tandem cold rolling mill, a two-dimensional finite element model with the assumption of plane strain conditions was developed using an explicit code. Due to symmetry, only half of the geometry was modeled. The rolls

y x

t/2

Rigid roll

Strip Symmetry Fig. 6. Finite element model for the rolling stand.

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100

Total reduction (%)

90

case 2

case 1

80

Unsafe Zone

70 60 case 4

case 3

50

Safe Zone Exp. data [12]

40

Selected cases

30

curve fit

20 1

1.5

2

2.5

3 3.5 Initial thickness (mm)

4

4.5

5

5.5

Fig. 7. Experimental diagram for strip tearing occurrence.

Table 5 Simulation results for selected cases.

Case Case Case Case

1 2 3 4

Total reduction (%)

Initial thickness (mm)

Maximum damage value

82 80 45 50

2 4 2 3

0.43 = Dcr 0.43 = Dcr 0.07 0.09

were defined as rigid surfaces, and the strip was modeled as a two-dimensional deformable solid using the elastic–plasticdamage properties that were described in Section 4. The model was discretized into a mesh that consisted of four-node plain strain elements. The total number of elements used in this study was 20,000 and remained the same for all simulations to minimize the effect of mesh dependency. A mixture of Coulomb’s law and constant-limit shear stress was used for the contact friction conditions. The values of friction coefficient at each stand were obtained from industrial reports [12]. Based on these reports, the rolling stands use different friction coefficients due to the variations in the surface conditions of the work rolls. The work rolls in the last stand are rougher for the purpose of surface finishing. Therefore, the friction coefficient for the fifth stand is the highest amongst all of the rolling stands. Fig. 6 shows a schematic diagram of the FE model for one of the rolling stands; the rolling direction and the through-thickness direction coincide with the x and y axis, respectively. The simulations were performed by using ABAQUS/explicit finite element code. To predict damage growth through the strip, the improved Lemaitre damage model was incorporated in the FE code by developing a user material subroutine (VUMAT) [21]. The efficiency of the proposed numerical model is discussed in the following section. 5.2. Validation of the FE model Validation of the efficiency and accuracy of the damage-coupled FE model was conducted using experimental data from the industrial rolling mill. An investigation of the industrial rolling programs, which were accompanied by strip tearing, revealed various factors that influence the tearing phenomenon, such as operator error and the defects in the input coils, as well as problems related to the mechanical and control equipment in the mill. In this section, the rolling setup properties, which include the total reduction percentage and the initial thickness of the strip, are considered as the dominant factors that govern the strip tearing phenomenon. Therefore, only the experimental strip tearing cases that are related to these factors were considered, whereas the other cases were ignored. The industrial rolling setups for which strip tearing occurred at Stand 5 (due to the primary factors) were selected and used to plot an experimental diagram for strip tearing occurrences, as shown in Fig. 7. This diagram shows the boundary between the safe and unsafe zones for possible occurrences of strip tearing at Stand 5. Strip tearing occurred at lower reduction ratios when the initial thickness of strip was larger. This experimental diagram can be used to verify the damage model for the prediction of damage evolution and strip tearing. The verification process was conducted by simulating different rolling setups and comparing the results to experimental observations. Four different rolling setups (Cases 1, 2, 3 and 4 in Fig. 7) were selected and numerically simulated. In this figure Case 1 lies on the curve fit, which indicates that it corresponds to an industrial rolling setup for which strip tearing occurred at Stand 5. Additionally, Case 2 is located in the unsafe zone, and thus strip tearing is expected as well. For Cases 3 and 4, which were selected from the safe zone, no strip tearing is expected. The simulation results for the four cases (besides the properties of each setup) are presented in Table 5.

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Fig. 8. Damage evolution through the strip for Case 1 at: (a) Stand 1, (b) Stand 2, (c) Stand 3, (d) Stand 4 and (e) Stand 5.

Table 6 The rolling setup properties and numerical results for Case 1.

Reduction (%) Angular velocity (rad/s) Friction coefficient Maximum damage

Stand 1

Stand 2

Stand 3

Stand 4

Stand 5

32 13 0.05 0.04

34 19 0.04 0.11

37 28 0.04 0.18

32 41 0.04 0.30

6 43 0.06 0.43

Crack initiation in a material corresponds to a critical value of damage, Dcr, which is equal to 0.43 for St14 steel. Therefore, it is clear from Table 5 that, according to the numerical simulation results, for Cases 1 and 2, the damage parameter reached its critical value at Stand 5, which resulted in strip tearing. For Cases 3 and 4, as expected, the damage parameter remained under the critical value, which indicates that strip tearing did not occur. In other words, the numerical predictions correspond to what was observed in practice because the numerical simulations predicted strip tearing for the cases that were located in the experimental unsafe zone. On the other hand, for the cases in the safe zone, no strip tearing was predicted by the numerical model. The simulation results for Case 1, which corresponds to an industrial rolling setup, are shown in Fig. 8. Due to space limitations, the results for each rolling stand are presented separately in this figure. Additionally, due to symmetry, only half of the strip thickness is shown. The properties of this rolling setup, including the reduction percentage, the friction coefficient and the angular velocity of the work rolls besides the maximum damage value for each rolling stand are shown in Table 6. Fig. 8a–e shows the damage distribution through the strip for each rolling stand. From this figure, it is clear that during the rolling process, the damage increases gradually until it reaches a maximum value at Stand 5. The pattern of the damage evolution during the process is shown in Fig. 9. As shown in Fig. 9, the damage evolution primarily occurs under the work rolls of each stand, and the damage value remains nearly constant in the inter-stand distances. The results indicate that the large plastic strains are the primary reason for damage evolution during the process, and the influence of inter-stand tension is negligible. An important physical aspect is that the numerical model predicts the formation of a special head-end shape for the final strip after exiting Stand 5, which corresponds to an end effect that is referred to as fish tail in practice. In Fig. 10, the numer-

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Fig. 9. Damage evolution pattern during the process.

Fig. 10. The end of the output strip according to the simulation results.

Table 7 Experimental and numerical roll force for each rolling stand.

Experimental roll force (kN) Numerical roll force (kN) Error (%)

Stand 1

Stand 2

Stand 3

Stand 4

Stand 5

8326 8060 3.2

9187 9500 3.4

5660 5726 1.2

6200 6464 4.2

7357 4800 34

ical prediction of the entire strip thickness for the final rolled strip end shape, after exiting Stand 5, is shown. Experimental observations show that the fish tail is usually occurred at the head end of workpiece. The end portion of the workpiece with the thickness deviations exceeding the acceptable tolerances is usually cut [12]. Additionally, the experimental and numerical values of the roll force are shown in Table 7 for each rolling stand. For Stands 1–4, the numerical results are in good agreement with the experiments; however, a considerable difference of 34% is observed for Stand 5. This difference stems from the ironing process that was conducted at the last rolling stand for the purpose of surface finishing in the industrial rolling mill. In fact, the force measured by the load cells at this stand is the sum of the rolling force

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0.45

Damage Parameter

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.07

0.09

0.11

0.13

0.15

0.17

Friction Coefficient Fig. 11. Variations in the maximum damage value versus the friction coefficient at Stand 5.

and the ironing force. Therefore, the experimentally measured force at the last stand is significantly larger than the numerical results, which represent the pure rolling force. In the following section, by employing the FE model, the influence of the effective factors for damage evolution and strip tearing will be investigated. 6. Effective factors for strip tearing Various factors can cause strip tearing in tandem rolling. In Section 5.1, the total reduction percentage and the initial thickness of the strip were considered as primary manageable factors to study the industrial strip tearing cases for verification purposes. However, other effective factors, such as defects in the input coils and problems related to the mechanical equipment in the mill and work rolls, are inevitable, and their influence on damage evolution through the strip should be investigated. Considering this goal, in this study, the effects of friction coefficient variations and the thickness difference between two welded strips are investigated by using numerical simulations. In addition, the reduction schedule is considered as another effective factor, and its influence on the damage evolution through the strip is investigated in this section. 6.1. Effect of friction coefficient variations The friction coefficient between the work rolls and the strip changes due to corrosion of the surface of the rolls or the use of different types of lubricant. The experimental data maintain that the friction coefficient in the cold rolling process is typically within a range of 0.05–0.15 [12]. In this section, the influence of friction coefficient variations on the damage evolution through the strip is studied. Numerical simulations are performed for different values of the friction coefficient, and the results are shown in Fig. 11. This figure shows the maximum damage value versus the friction coefficient at Stand 5. This figure also shows that the maximum damage value increased along with an increase in the friction coefficient until it reached a critical value, Dcr = 0.43, for f5 = 0.15, which resulted in strip tearing at Stand 5. Table 8 presents values of the friction coefficient at the rolling stands for four different experimental friction conditions. The distribution of damage through the strip at Stand 5 is shown for Cases 1 and 4 in Fig. 12. Because of symmetry, only half of the strip thickness is shown in this figure. Fig. 12 shows that for small values of the friction coefficient (Case 1), the maximum damage was detected at the strip surface. An increase in the friction coefficient caused the maximum damage area to move towards the center of the strip and localize there (Case 4) because an increase in friction causes the plastic deformations to penetrate the thickness of the strip. The results indicate that the friction coefficient has a significant effect on damage evolution through the strip. The amount of damage increases along with an increase in the coefficient of friction due to an increase in deformation inhomogeneity and an increase in plastic strain. The critical value for the friction coefficient, which had a maximum value in the last stand, is equal to 0.15; therefore, the recommended deadline for replacing the work rolls is when the friction coefficient reaches this critical value due to corrosion and abrasion on the rollers. Therefore, although friction is an essential part of the rolling process in practice, it is necessary to limit the amount of friction to reduce the probability of strip tearing in tandem cold rolling. 6.2. Effect of thickness difference between two welded strips According to industrial reports, in some cases, to provide continuous feeding and avoid stopping the mill, two input coil strips with different thicknesses are welded to each other, which results in a sudden change in the entry strip thickness. Gen-

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M. Mashayekhi et al. / Simulation Modelling Practice and Theory 19 (2011) 612–625 Table 8 Friction coefficient values for four different friction conditions [12].

Case Case Case Case

1 2 3 4

Stand 1

Stand 2

Stand 3

Stand 4

Stand 5

0.06 0.09 0.14 0.16

0.05 0.08 0.13 0.15

0.05 0.08 0.13 0.15

0.05 0.08 0.13 0.15

0.07 0.10 0.15 0.17

Fig. 12. Damage distribution through the strip at Stand 5 for different friction conditions: (a) Case 1: f5 = 0.07, maximum damage zone (D = 0.21) was detected at the strip surface, (b) Case 4: f5 = 0.17, maximum damage zone (D = 0.43) was detected at the center of the strip.

erally, a change in the thickness of the input strip should be accompanied by appropriate changes in the controlling systems. However, because of the high speed of the process, it is not possible to make immediate changes. Therefore, the system would encounter a sudden increase in the entry strip thickness, which can result in an increase in the rolling force and consequently in the parameters that are related to the rolling force, such as stress and inter-stand tension. Consequently, the thickness difference between the two welded strips can be considered to be another effective factor related to damage evolution through the strip, and its influence is studied in this section. Numerical simulations were performed for four different industrial cases with the thickness differences of 10%, 16%, 30% and 37% with a basic strip thickness of 2.5 mm [12]. Fig. 13 shows the FE model and the numerical results for a thickness difference of 16%. Fig. 14 illustrates the damage evolution during the process for each thickness difference percentage. It is clear from this figure that the maximum damage value increases slightly along with an increase in the thickness difference, and the damage evolution pattern is similar for all of the cases. In general, this parameter cannot be considered to be a crucial factor for a strip tearing occurrence because it does not significantly affect the magnitude and evolution rate of damage. 6.3. Effect of reduction schedule In the tandem rolling process, various reduction schedules can be employed for desired values of the total reduction ratio and initial strip thickness. Many attempts have been made to propose reduction schedules to optimize the operation of tandem mills with respect to the mill throughput, power utilization or other dependent parameters [22]. In Reddy and Suryanarayana [22], for a five-stand tandem mill, the power consumption is calculated and compared for various reduction

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Fig. 13. (a) FE model and (b) damage distribution through the strip at Stand 5 for a thickness difference of 16%.

0.3

Damage Parameter

0.25 0.2 10% thickness difference

0.15

16% thickness difference 30% thickness difference

0.1

37% thickness difference

0.05 0 0

1

2

3

4

5

6

Stand No. Fig. 14. Damage evolution for various industrial cases of thickness difference.

schedules by distributing the strip thickness in arithmetic, geometric and harmonic series and for the reduction schedule proposed by Roberts [23]. In this study, these reduction schedules are investigated and compared from the continuum damage mechanics point of view. The arithmetic, geometric, harmonic and Roberts reduction schedules were numerically simulated and compared with respect to the damage evolution through the strip during the rolling process. The simulations were conducted for a special case in which the initial and final thicknesses of the strip were 2 mm and 0.6 mm, respectively, which corresponds to a total reduction of 70%. Table 9 shows the reduction percentage of each rolling stand, Ri, for the rolling schedules; Ri is obtained from the following relation:

Ri ð%Þ ¼

hin  hout  100; hin

ð12Þ

where i represents the stand number and hin and hout refer to the entry and exit strip thicknesses for that rolling stand, respectively. The value of the damage parameter at each stand, which was obtained from the numerical simulations, is shown in Fig. 15 for each rolling schedule. It is clear that the damage value at each stand attains its minimum value if the strip thickness is distributed in an arithmetic series. For this case, the maximum damage value is Dmax = 0.15 at Stand 5.

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M. Mashayekhi et al. / Simulation Modelling Practice and Theory 19 (2011) 612–625 Table 9 Percentage reduction schedules for a five-stand mill [22]. Stand no.

Roberts

Harmonic

Geometric

Arithmetic

1 2 3 4 5

32.50 29.62 23.15 15.06 1.61

31.35 23.81 19.31 16.11 13.84

21.15 21.17 21.15 21.12 21.08

13.90 16.14 19.25 23.84 31.30

0.3

Damage Parameter

0.25 0.2

Geometric Harmonic

0.15

Arithmetic Roberts

0.1 0.05 0 0

1

2

3

4

5

6

Stand No. Fig. 15. Damage parameter at each stand for different reduction schedules.

On the other hand, for the Roberts reduction schedule, the damage value at each stand attains its maximum value among the reduction schedules, and Dmax = 0.24 at Stand 5. Therefore, the selected reduction schedule affects the damage evolution during the process, and for the special values of total reduction and strip initial thickness, the reduction schedule obtained by distributing the strip thickness in an arithmetic series gives rise to the minimum damage evolution through the strip. 7. Conclusions This paper investigated the strip tearing phenomenon in a tandem cold rolling process by employing FEM besides CDM. The industrial five-stand tandem cold rolling mill at IMSC was simulated using an explicit finite element code. The damage evolution through the strip was predicted by applying the improved Lemaitre damage model that was incorporated in the FE code. The simulation results were in good agreement with experimental observations, which verifies the correctness of the proposed damage-coupled finite element model. The influence of some operating factors such as the friction coefficient, thickness difference between two welded strips and reduction schedule on the damage evolution through the strip were investigated by using numerical simulations. The numerical results indicate that the friction coefficient is a dominant factor in damage distribution through the strip because the damage increases along with an increase in the friction coefficient. Additionally, friction conditions can affect the damage initiation area in the strip. Therefore, it is necessary to carefully consider the lubrication and surface corrosion of the rolls to maintain the friction coefficient at the lowest possible level to reduce the probability of strip tearing in tandem cold rolling. In addition, the thickness difference between two welded strips does not significantly affect the damage evolution pattern, and there is no significant increase in the maximum damage value due to an increase in the thickness difference. The results indicate that the reduction schedule is an effective factor in damage evolution, and it is necessary to select a proper reduction schedule to reduce damage in the process. Based on the results of this study, the amount of damage can be minimized by using a reduction schedule that is obtained by distributing the strip thickness in an arithmetic series. In this study, a fully isotropic material behavior and an isotropic damage process were taken into consideration. However, developing more accurate equations accounting for the initial and induced anisotropy remains a challenge for future development work. References [1] W. Johnson, A.G. Mamalis, A survey of some physical defects arising in metal forming processes, in: Proceeding of the 17th International MTDR Conference, Macmilla, 1977, p. 607.

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