4.1. Analysis and Results of the Analytical Model
Figure 6 shows the stress–strain curves of a mild steel under a uniaxial, symmetric strain-controlled, cyclic test. The strain range
is 0.015. Based on the first half of the stress–strain curve of the first cycle, the calibrated values of the Young’s modulus
and the initial yield stress
are 219.8 GPa and 329.7 MPa, respectively, and
and
have values of 66.3 MPa and 742.7, respectively. The values of
and
are calibrated as 128.3 MPa and 604.7, respectively, based on the second half of the stress–strain curve of the first cycle.
By specifying the combined hardening coefficient
, computational stress–strain curves based on the analytical model of Equations (1)–(11) with the strain range
of 0.0015 can be obtained. For a nine-roll roller leveler, the total number of bends equals seven. The values of the combined hardening coefficient
for the seven bends can be calibrated by comparing the computational stress–strain curves with the stress–strain curves of the mild steel shown in
Figure 6 for the first seven bend loadings.
Figure 7 shows the computational stress–strain curves and the stress–strain curves of
Figure 6. The computational curves agree with those of the mild steel. The calibrated
values for the second to the seventh bending are 0.0784, 0.2, 0.3, 0.4, 0.45, and 0.5, respectively. The hardening behavior of the material for the first bending is taken as isotropic.
Consider the case where the strip has the thickness (=20 mm), the length (=2480 mm), and the width (=100 mm). The radius of the rolls is 85 mm. The roll spacing is 150 mm. The strip is taken as flat and free of residual stress. The upper roll carriage has an inclination angle of 0.1° and the roll intermesh at the next to the last roll ranges from −1.2 mm to 0.8 mm. Note that the value of the roll intermesh is positive when the gap between the top roll and the bottom roll is larger than the strip thickness. Therefore, a negative value of the intermesh means the work roll plunges into the strip.
With
= −0.70 mm and the upper roll carriage inclined at the angle
of 0.1°,
,
, and
have values of −2.32 mm, −1.78 mm, and −1.24 mm, respectively. Sixty divisions are taken along the thickness direction of the strip for calculation of the stress and strain distributions.
Figure 8 shows the deformed strip between roll 1 and roll 9 based on the analytical model.
Figure 9a–c shows the distribution of the bending moment per unit width, the curvature, and the deformed center line of the strip, respectively, between roll 1 and roll 9. The contact points between the strip and the rolls are marked by circles in the figure. One hundred nodes are used between the contact points in the computations.
Figure 9a shows the linear distribution of the bending moment between the contact points. At the entry roll and the exit roll, the values of the bending moment are zero.
Figure 9b shows the curvature distribution. Values of the curvatures are nearly two orders smaller than the roll curvature
. The curvature of the strip at the exit roll has a value of
. The positive value of the curvature indicates that the strip is bent upward. Higo et al. [
10] pointed out that the abrupt increase in magnitude of the curvature near all contact points except the final one is due to the nonlinear material hardening behavior. This is evidenced by the nonlinear sections of the moment–curvature curves during the leveling process plotted in
Figure 10. The moment and the curvature are normalized by the maximum bending moment
and the maximum curvature
for which elastic conditions hold, respectively.
Figure 11a–c shows the stress distributions in the strip thickness direction when the strip travels through the leveler. The results computed by the analytical model are displayed by lines in the figure. Significant plastic deformations appear in the first three bends as seen in
Figure 11a. The fractions of the plastic deformation are nearly 40%, 50%, and 60% for the first, the second, and the third bend, respectively. As the strip traverses through the fifth roll and the sixth roll, the region of plastic deformation stays at 60% as seen in
Figure 11b.
Figure 11c shows the stress distributions in the strip thickness direction at the eighth roll and the exit roll. The stress distribution at the exit roll can be taken as the residual stress distribution since the total moment applied to the strip at the exit roll is nearly zero. The stress at the
th roll is calculated based on a linear superposition assumption. The remnant stress of the cross section of the strip at the
th roll and the loading stress at the
th roll are superimposed to obtain the stress distribution at the
th roll. Guan et al. [
25] also adopted this stress inheritance law in their roller leveling model. Yonetani [
26] reported that the stress of a microscopic segment at the cross section in a uniaxial stress state satisfies the linear superposition assumption. The uniaxial stress loading condition is also assumed in the model considered in this investigation.
4.3. Residual Curvature
Flatness is an important factor to evaluate the strip quality after roller leveling processes. The residual curvature of the strip at the exit roll can be used as a metric to evaluate the strip flatness after leveling.
Figure 13 shows the residual curvatures
of the strip as a function of the averaged intermesh based on the analytical model and the finite element analyses. The averaged intermesh is the averaged value of the roll intermeshes,
,
,
, and
, where
ranges from −1.2 mm to 0.8 mm. The residual curvature
is an indicator of the deviation of the strip from an initially flat surface at the entry roll. The curve with the dash-dot line and the markers represent the results based on the analytical model and the finite element analyses, respectively. The analytical predictions generally agree with the finite element analyses for the averaged intermeshes ranging from −1.71 mm to 0. As the averaged intermesh decreases from zero, the values of
decreases.
appears to exhibit an oscillatory behavior when the averaged intermesh is less than −0.11 mm based on the analytical model. As the averaged intermesh decreases further, the amplitude of the oscillation of
grows. When the averaged intermesh is less than −1.41 mm (analytical predictions),
has positive values of increasing magnitude. Five crossover points are observed at the averaged intermesh of −0.36 mm, −0.61 mm, −0.86 mm, −1.11 mm, and −1.41 mm (analytical predictions). Smith [
27] reported that the several crossover points with zero residual curvature underlie the reason why successful leveling can be achieved by the series roll leveling process in practice.
In the analytical model, a point contact is assumed between the strip and each roll, which means the strip does not wind around the work rolls. This contradicts with the fact that multi-point contact between the strip and the roll predicted in the finite element analyses. Morris et al. [
3] reported that the wrap angle near the exit roll has a significant influence on the flatness of the strip. Wrap-around contact length between the strip and the roll may depend on the intermesh and roll spacing. In describing the arc of contact of the strip around a roll, an effective radius can be assumed to model the wrap-around contact characteristics. The concept of the effective radius is illustrated schematically in
Figure 14.
Figure 14a shows the original contact model, where the strip contacts with roll
tangentially.
is the expanded radius of the roll, which is defined by Equation (7). (
,
) is the
th contact point, and
is the
th contact angle.
Figure 14b shows that the strip makes contact elastically with roll
when subjected to an external force
. The circumferences of the deformed roll and the original roll are represented by the solid line and the dashed line, respectively, in
Figure 14b. (
,
) is the
th contact point, and
is the
th contact angle for the Hertz contact model. A local deformation is ensued to cause a reduction in the local radius of roll
. The effective radius of the roll
is given as
where
is the contact compliance of roll
. The external force
is computed from the moment distribution in the analytical model. Based on the Hertz contact model, the contact point
is
where
is the coordinates of the center of the roll. As shown in
Figure 14b, the deformed center line of the strip based on the analytical model with the Hertz contact compliance may be thought to have an arc segment bounded by the two virtual contact points,
and
, wrapped around the circumference of the original roll
, shown as a dashed circle in
Figure 14b. When the contact angle
is very small which is the case in the roller leveler, the line segment and the arc segment bounded by the two end points
and
are approximately equal. The contact compliance of the rolls is taken as a fitting parameter in the analytical model with the Hertz contact compliance to fit the analytical predictions with the finite element computations. Yi et al. [
28] considered the wrap angle on a roll during a roller leveling process by fitting an arc curve around a roll. A parameter determined by experiments is needed for curve fitting.
The residual curvatures
of the strip calculated by the analytical model with the Hertz contact compliance are plotted in
Figure 13. The curves with the compliance
= 0 and
mm/N appear to envelop the results based on the finite element analyses. The curve with the compliance
=
mm/N may be able to predict the general trend of the results based on the finite element analyses for the averaged intermesh ranging from 0 mm to −2 mm. The wrap-around contact condition between the strip and the rolls can be manifested by the analytical model with the Hertz contact compliance within an acceptable accuracy, compared with the results of the finite element analyses.
A series of simulations was run to calculate the residual curvatures
of the strip with an initial curvature
based on the analytical model with the Hertz contact compliance
=
mm/N. The initial stress in the strip is neglected without losing generality of the residual curvature predictions. Mathieu et al. [
29] considered initial flatness defects in their finite element analyses of a leveling process. They introduced the flatness defects in the strip which was free of stress.
Figure 15 shows the residual curvature as a function of the averaged intermesh for the strip with the initial curvature
varying between
to
. As shown in
Figure 15, at low levels of roll intermesh (averaged intermeshes greater than −0.1), the residual curvatures
for the three cases of
=
,
, and
deviate from each other significantly. For the averaged intermesh in this level, the curve for the case of
=
oscillates mostly in the positive-curvature region (
> 0), in contrast to the cases of
=
and
, which oscillate between the positive
region and the negative
region. The initially bowed-down defect for the strip with
=
exits the leveler with the residual curvature in the same direction for the averaged intermesh greater than −0.71. As the extent of the averaged intermesh increases, the residual curvatures for the three cases of
=
,
, and
gradually converge to the same values. For the values of the averaged intermesh less than −1.1, the curves for the three cases appear identical, where two cross over points with the interpolated averaged intermesh values of −1.46 and −1.13 were found. For the values of the averaged intermesh less than −1.46, positive residual curvatures were produced. For the averaged intermesh values within the interval of −1.46, −1.13, the minimum of the residual curvature is
. In this region of the leveler settings, the residual curvatures of the strip seem to be insensitive to its initial curvatures. This result can serve to the advantage of leveler operators to obtain nearly zero residual curvature for strips with various initial curvatures. Grüber et al. [
18] also demonstrated robustness of the roll intermesh settings for a roller leveler regarding a change in the initial curvature. In this investigation, the plane strain condition is considered in the finite element analyses. The width-to-thickness ratio of the strip considered in the analyses is 5. Carvalho et al. [
30] reported that, in order to develop near plain strain conditions, it is important to maintain a ratio of width-to-thickness greater than 5.
A combined isotropic/kinematic hardening is implemented to describe the material hardening of the strip in this investigation. Doege et al. [
5] also adopted combined isotropic/kinematic hardening for their leveling model, where mathematical formulations of their hardening model were not presented. Detailed formulations of the hardening model are provided in this investigation. Doege et al. [
5] presented analysis results of their model. Results of the stress distributions and residual curvature of a steel strip based on our analytical model are verified by the finite element analyses. Doege et al. [
5] computed the contact points between the strip and the rolls by assuming only one contact point between the strip and each roll. An effective radius modelling the wrap-around contact characteristics by the Hertz contact compliance is proposed to describe the arc of contact of the strip around a roll. A roll inter-mesh range to produce a flatness condition of the strip is presented based on the analytical model with the Hertz contact compliance.
Indeed, the initial curvature considered in the model is the longitudinal wave defect of a strip. Behrens et al. [
6] sectioned a strip longitudinally and showed that the length of all sections after leveling should be the same in order to remove a transverse wave defect. Therefore, bendable rolls, as practiced in the industry, can be applied into a leveler to cause various degree of plastic deformation in each longitudinal section to achieve equal length. Chen et al. [
9] developed an analytical model of a roller leveler with consideration to the bending of the rollers to eliminate transverse wave defects. This approach can be implemented in the analytical model to extend its applicability. Park and Hwang [
13] slit a strip longitudinally to calculate the initial curvature of each longitudinal section. Given proper roll intermesh settings, the longitudinal sections with various initial curvatures can reach similar values of residual curvature after leveling based on finite element analyses and experiments. The results shown in
Figure 15 based on the developed model also provide evidence that the residual curvatures of the strips with different initial curvatures can converge to the same value given enough amount of roll intermesh.
Given the multiple forming processes and complex machine settings involved in the roller leveling, tradeoffs between a simple, efficient model and an elaborate, detailed model should be balanced. Baumgart et al. [
31] described that the effects of parts of the leveler, such as support rolls, frames, posts, and adjustment screws, should also be considered in order to obtain a more accurate leveler model. Wang and Li [
32] reported that stiffness of roll cassettes and leveler housing are important factors of the leveling process. In this study, a relatively simple analytical model was developed based on the assumptions of two-dimensional geometry, pure bending of the strip, and the uniaxial loading condition. Compared to previously reported models, the Hertz contact compliance implemented in the model relaxed the single point contact condition between the strip and the roll, and a relatively accurate prediction of the residual curvature can be attained. The model could serve as a guide in the development of strategies for effective adjustment of roller levelers.