2.2.1. The Biting Angle and the Length of Contact Arc
Since dynamic adjustment of the roll gap is required during VGR, the forming theory of BVTs-RD is quite different from that of common rolling. Various complex shapes of LP plates and TRB plates can all be considered as a combination of downwards rolling (DR), upwards rolling (UR), and common rolling (CR) over one or multiple stages [
28]. In comparison to the conventional rolling process, the rolling process of the wedge section on the longitudinal variable cross-section leads to changes in the exit position of the workpiece, as shown in
Figure 3.
T refers to friction force,
P to positive pressure, and α to the biting angle. Ignoring the elastic spring-back deformation of the workpiece at the exit and the elastic flattening of the rolls, the transition area rolling has the following characteristics. The line connecting the exit position with the center of the roll must be perpendicular to the inclined surface of the wedge section. The angle between the line connecting the point where the exit position contacts the roll with the center of the roll and the center line of the roll is equal to the inclination angle
φ of the wedge section. The inclination angle
φ of each wedge section is constant, so the exit position of the workpiece on the roll remains unchanged. With the rolling of the wedge section, the forward slip value and the exit speed of the workpiece continuously change [
29].
The initial bite-in process of VGR is the same as simple rolling. Du [
29] believes that
α in both upward rolling and downward rolling is consistent with that in conventional rolling, meaning that the bite angle α remains constant during the rolling process. As the rolling process continues, the angle
Φ between the center line of the roll and the resultant force will continuously change. For upwards rolling,
Φ decreases gradually from the initial
α. Once the metal fills the rolls,
Φ becomes (
α +
φ)/2. For downwards rolling,
Φ decreases gradually from the initial
α. Once the metal fills the rolls,
Φ becomes (
α −
φ)/2. Zhang [
30] also pointed out that the friction conditions required for the bite-in of upwards rolling are the highest. For downwards rolling, when the inclination angle is too large, there may be situations where the bite-in can occur smoothly while the rolling process cannot. Dong [
31] believes that the bite angle in variable thickness rolling differs from that in conventional rolling. It is not only related to the reduction ∆
h and the roll radius R but also to the inclination angle
φ of the wedge section. Sun [
32] established a functional relationship between
α and the vertical movement speed of the roll and time. This expression offers improved accuracy, but it increases the difficulty of solving some rolling parameters involving
α because time is used as a variable. Considering the calculation accuracy of the bite angle, the latter two expressions are more accurate. However, from the perspective of engineering applications and subsequent dynamic setting of the rolls, the research results of the former two are more convenient to apply. Specific bite-in conditions and continuous rolling conditions are detailed in
Table 1.
The deviation of the exit point in VGR leads to a corresponding change in the contact arc length. Based on the geometric characteristics of the deformation zone in VGR, as shown in
Figure 3, the expressions for the contact arc length in UR and DR can be easily derived [
29,
30,
33]:
where
lup and
ldown are the contact arc length for UR and DR, respectively;
R is the roller diameter;
α0 is the wedge angle of the workpiece; and ∆
h is the reduction.
x0 is the distance from the exit to the centerline of the roll. Dong [
31] believes that ∆
h/2 in the above formula is a relatively small quantity that can be neglected, thus obtaining an expression for a contact arc length that is more conducive to calculation. Generally speaking, the reduction is relatively important in the calculation of rolling parameters such as the rolling force, and its value has a significant impact on the calculation results. Given that, neglecting the reduction is not an appropriate simplification method, although it may provide convenience in engineering applications and further dynamic control of rolls.
2.2.2. The Forward Slip Formula
The presence of an inclination angle in the longitudinal VGR process changes the action point of the workpiece leaving the rolls. The conventional formula for forward slip cannot meet the requirements of the longitudinal variable section rolling process. However, the definition formula
f = (
vφ −
v0)/
v0 still holds, in which
v0 represents the linear velocity and
vφ represents the horizontal exit velocity of the workpiece. The metal flow velocity of the section in the deformation zone which corresponds to the neutral angle is
vγ =
v0cos
γ. According to the constant mass flow principle,
vγhγ =
vφhφ,
hγ can be deduced, where
hγ is the thickness of the workpiece at the neutral plane and
hφ is the thickness of the workpiece at the exit [
28,
29,
30,
34]. For the UR process shown in
Figure 3a, the geometric relationship
hγ −
hφ =
D(cos
φ − cos
γ) holds. Based on the above conditions, the formula for forward slip during UR can be derived [
28]:
The formula above also applies to the calculation of forward slip during DR. With
hφ and
γ in downwards rolling being substituted in, the forward slip value can be obtained. This method is derived based on the definition of forward slip, combined with the geometric relationship between the neutral plane thickness and the exit section thickness in the deformation zone. Using this forward slip calculation method, assuming the entire contact surface is fully sliding and follows Coulomb’s friction law in the VGR process, and based on the conditions of equal front and back tension, as well as equal unit pressure at the neutral plane, Yu [
35] derived the neutral plane thickness.
where
h is the thickness of the workpiece when entering the roll gap and
μ is the friction coefficient of the rolling process. In order not to solve
γ,
hγ −
hφ = d(cos
φ − cos
γ) is substituted into Equation (3) to yield the relationship between the forward slip value and neutral plane thickness.
Equation (4) can be substituted into Equation (5) to calculate the forward slip value.
Another idea is that the metal flowing out of the exit section consists of two parts: the metal flowing into the entry section and the metal flowing from the deformation zone due to the vertical movement of the rolls. Based on this mathematical relationship, using the average velocity and thickness of the metal at the neutral plane to express the velocity of the metal at the exit section, and substituting the roll linear velocity into the expression for the forward slip coefficient, another expression for the forward slip value can be derived [
31]:
where
λ = −1 is for UR,
λ = +1 is for DR, and
vy is the vertical velocity of the roll.
The first method for calculating forward slip has been applied to industrial production and its accuracy and effectiveness have been verified through actual on-site production. The second calculation method is an improved algorithm based on the first one, with the advantage of avoiding complex solutions for the neutral angle and instead representing the forward slip value using the neutral plane thickness. However, while the neutral plane thickness is easy to solve, its representation is relatively cumbersome, making it inconvenient for industrial applications. Different from the first two methods, the third method uses velocity as an important parameter to represent the forward slip value. This results in an increased number of parameters, but the significant advantage is that the parameters in the forward slip expression are all simple ones that can be obtained without complex solutions. Although this method has not been applied in industrial production, the author believes that it may receive attention in the future due to its ease of application.
2.2.3. The Force Equilibrium Differential Equations
One important aspect of studying Variable Thickness Rolling is statics analysis. A new force equilibrium equation different from the conventional rolling process was established to derive the calculation formula for rolling force applicable to VGR and provide guidance for the practice of VGR. In the analysis of the VGR process, two assumptions are specifically proposed in addition to the assumption adopted when deriving the Karman Equation [
36]. One is that the transition area curve of the workpiece is uniformly continuous, and the other is that the rigid displacement speed of the roll uniformly and continuously changes. Here, the rigid displacement speed refers to the motion speed perpendicular to the rolling direction at the center of roll rotation [
23,
30,
37,
38,
39]. The unified approach in the literature is to utilize the symmetry in the vertical direction, and half of the elemental with a width of
dx in the deformation zone is analyzed, as shown in
Figure 4.
Like conventional rolling, this method is based on the Karman equilibrium differential equation. It is reasonable to take the contact arc between the rolled piece and the roll as a straight line. However, there is also a hidden condition—the shape of the transition zone is assumed to be the straight line type, and also the simplest one. If the curve type changes, the equilibrium differential equation established by this method will no longer be accurate. During upwards rolling, the roll is lifted a distance of
δy in an interval of ∆
t, causing the line unit ED to become AC at time
t + ∆
t. A thickness of 2(
y +
δy) for the workpiece is obtained. From the geometric relationship in
Figure 5, it can be inferred that [
37]
where
ϕ is the wedge angle during conventional rolling and
ϕ′ is that during upwards rolling. Assuming the lifting velocity of the roll during upwards rolling is
vy, and the horizontal velocity is
vx, then it follows that
Substituting Equations (7) and (9) into Equation (8) yields
A unit width of strip steel was measured to be studied. On the AC line of the elemental body, there is a horizontal tensile stress σx, with a resultant force of . On the ED line, there is a horizontal tensile stress σx + dσx, with a resultant force of .
At the contact surface, is valid in the backward slip zone, and is valid in the forward slip zone.
According to the force equilibrium condition, the resultant force in the horizontal direction acting on the elemental body is [
37]
The above formulas are rearranged and simplified to
where + indicates that the element is located in the forward slip zone, and
indicates that the element is located in the backward slip zone. It can be seen that during the upward rolling, the force balance equation and Karman’s equation are essentially the same. With only the changes in the micro-element shape, the equation can be obtained according to the new geometric relations.
The force equilibrium differential equations can be established in downwards rolling if a similar analysis method as upwards rolling is employed. The difference is that during downwards rolling, the workpiece will deviate towards the exit side when it leaves the contact point with the rolls, resulting in a deformation zone slightly larger than that of the simple rolling process, as shown in
Figure 5. The contact arc tangent in the deformation zone of UR changes in the opposite slope, resulting in the non-existence of an equilibrium equation applicable to the whole deformation zone. In the analysis, it is necessary to divide the deformation zone into Zone I and Zone II along the centerline of the two rolls for separate study.
For the elements in Zone I, using the force equilibrium condition, the same force equilibrium equation as in upwards rolling can be obtained, with the only difference being that the size of
is not equal. For downwards rolling,
. For the elements in Zone II, the contact arc length from the centerline of the two rolls to the exit can be approximated as ∆
l =
Rθ. It can be noted that the elements in Zone II are usually in the forward slip zone. Using the force equilibrium condition, we can obtain [
39]
By establishing and solving the force equilibrium equations, the distribution of unit pressure in the deformation zone can be obtained. The formula for calculating the rolling force can then be derived. Zhang [
39] established the equilibrium equation for the micro element in the deformation zone in the vertical direction:
where
σy represents the stress in the vertical direction.
σy = −
p can be obtained because
is very small. Considering the deformation as plane deformation, we can obtain
σx = 2
k −
p, where 2
k = 1.155
σs. Taking into account the strain hardening in cold rolling, and neglecting the relatively small terms in the equation, Equation (13) can be rewritten as
To sum up, Karman’s equation can be regarded as a special case when the roll speed in the force equilibrium differential equation is zero during Variable Thickness Rolling. It is worth noting that the force equilibrium equation is not continuous at the center line of the roll, so it may affect the calculation of force parameters such as the rolling force and even the precision of gap setting. Since the TRB has been produced stably, the above-mentioned problems may be ignored, or better mathematical equations may be constructed in the future.
2.2.4. Horizontal and Vertical Velocity Field
The shape factor tan
ϕ′ in the deformation area in the force equilibrium equation is the most important parameter in the equation. To determine its magnitude, a kinematic analysis of the VGR must be carried out. The shape of the transition area determines the relationship between the speed of the roll lifting and the actual horizontal rolling speed at the exit point of the rolling, while the shape of the deformation area determines the speed distribution of the workpiece throughout the deformation area. Only when the above relationships and speed distributions are made clear can the speed system be established and roll gap settings be made [
38,
40].
The transition area curve is typically represented by a cubic function. For this type of transition area, Fang [
41] derived the expression for the transition area curve from the thin area to the thick area.
where
H is the thickness of the thick area,
h is that of the thin area, and
l is that of the transition area, as shown in
Figure 6.
Based on this, considering that only when the vertical movement of the rolls matches the horizontal movement of the strip can the target transition area curve be achieved, the author derived the formula for the horizontal velocity at the exit and the vertical velocity of the rolls in the transition area rolling.
During the rolling of the transition area, the entire area is considered as the deformation area. Therefore, the metal undergoing deformation follows the principle of equal flow rate, which can be expressed as
where
C is the constant of the flow rate,
Vx is the rolling speed at the entrance of the thin area, and
hx is the thickness of the thin area. At the point (
x,
f(
x)),
Substituting Equation (18) into Equation (20) yields [
41]
Since the ratio of the vertical speed of the rolls at this point to the horizontal rolling speed is the slope, then
Equation (21) represents the expression for the horizontal speed at the exit point during UR, and Equation (22) represents the expression for the lifting speed of the rolls. Under the condition of DR, the equation for the cubic curve in the transition area from the thick area to the thin area is as follows:
Similarly, the expression for the horizontal speed at the exit point during DR can be obtained:
The expression for the vertical speed of the rolls is
When the transition zone curve type is known to be a cubic curve, by using the above formula, the velocity of any point on the curve and the relationship between the horizontal velocity and the vertical velocity of the roll can be calculated; however, the unit pressure distribution in the transition zone can still not be obtained without the horizontal velocity distribution in the deformation zone.
Based on the principle of volume invariance, Zhang [
42,
43] further studied the horizontal velocity distribution of metal in the deformation zone. The upward rolling is taken, as shown in
Figure 7. At time
t, the horizontal velocity of the micro element at BC (x) is
vx, and at AD (
x + d
x) it is
vx + d
vx. Due to the lifting of the rolls, at time
t + ∆
t, the thickness of the workpiece at the positions
x and
x + d
x changes from BC and AD to B′C and A′D, respectively. The horizontal velocity of the rolled piece during UR is a function of time
t and position coordinate
x. According to the condition of constant volume, it can be known that the metal flowing from the cross-section of the workpiece from the coordinate
x + d
x within time ∆
t consists of two parts: one is the metal flowing out from the cross-section at the coordinate
x, and the other is the metal enclosed by AA′B′B in
Figure 7. It can be expressed as
Assuming the vertical speed of the roll is
vy, the volume of the workpiece flowing into the cross-section at
x + d
x within time ∆
t can be obtained through integration:
Similarly, the volume of the workpiece flowing out of the cross-section at
x within time ∆
t can be obtained:
The volume enclosed by AA′BB′ can be expressed as
By combining Equations (26)–(29), we can obtain
Let the contact arc function at time
t0 be
, substituting it into Equation (30) and simplifying, we can obtain
The general solution of Equation (31) is [
42]
Assuming that the UR starts at time 0, after a time of
t0, the coordinate of the exit in the deformation area is
x =
Rsin
θ, the horizontal velocity of the exit of the workpiece is
vx =
vh1, and at the exit
.
θ is the inclination angle of the wedge section,
vh1 is the actual horizontal velocity of the workpiece exit, which can be obtained through the speed roll at the exit, and
h1 is the actual exit thickness at time 0. After the UR for time
t0, the exit thickness satisfies
Substituting Equation (33) into Equation (32) and simplifying, we obtain
Substituting Equation (34) into Equation (32), and letting ∆
t approach 0, we obtain
The analysis method of DR is the same as that of UR. By derivation, the horizontal speed of the workpiece during DR can be obtained as [
42]
Formulas (21) and (22), as the earliest velocity analytical formulas in the VGR process, lay a foundation for follow-up research and provide strong support for the industrial application of the VGR process. But they have their drawbacks. First, the calculation result is obtained based on the assumption that the equal discharge per second condition of each section in the deformation region holds. However, after analyzing Equation (35), it can be known that the second flow in the VGR process is a function of the coordinate
x, showing that the second flow of each section is not equal at the same time. For UR, the second flow decreases gradually from the entrance to the exit. In addition, the research results of Fang [
41] are based on the fact that the curve in the transition zone is a cubic curve. If the curve type changes, the relationship between horizontal and vertical velocities changes accordingly, resulting in the change in the final velocity formula. Thus, Equation (35) is a more reasonable and accurate velocity formula.
In addition, Zhang [
44] discretized the VGR transition zone and simulated the rolling process. Based on the simulation results, the author determined the rolling speeds for the thick area and the thin area based on the rolling theory of the constant thickness area. They represented the rolling speed in the transition area simply as the average speed of the constant thickness area rolling.
where
is the rolling speed in the transition area,
is that in the thin area, and
is that in the thick area. The rolling time in the transition area can be obtained using the following equation:
where
is the length of the transition area. Since the vertical velocity in the transition area can be expressed as
, substituting Equations (37) and (38) into this expression gives [
44]
This derivation method is simple and practical, offering new ideas and theoretical guidance for the vertical motion control of rolls. But, due to significant simplifications and the improper simplification method of horizontal velocity, the calculation accuracy can be greatly affected. Although the validity of the mathematical model has been verified by finite element simulation, the lack of experimental verification makes the model less convincing. In addition, the author also over-looked a key point: the transition area shape was not considered.
The establishment of the above velocity field expression is based on the curve function often used in actual production. Wang et al. [
45] put forward a double power function used for the transitional curve, which is continuous and smooth at all connection points, independent of its parameters, thus avoiding sudden changes in mechanical parameters during the rolling and forming process, as shown in
Figure 8.
The authors established the horizontal velocity field of the rolled piece and the vertical velocity field of the roll suitable for this transitional curve [
45]:
where
γ is the neutral angle;
vr is the roller′s rotational velocity; and
hγ is the workpiece′s thickness at the neutral point.
hx is the workpiece′s thickness at the point of tangency.
The authors also pointed out the limitations of this velocity field expression, as it is not applicable to all transitional curve profiles. In order to ensure the validity of the formula, the local coordinate of the roll’s lowest point in the η direction must be less than half the difference between h1 and h2 until ξ = L. Here, h1 and h2 represent the thickness of the thick and thin areas, respectively, while L is the length of the transitional area.
At present, there are four different research results about the velocity in the deformation zone during the VGR process. It is obvious that the shape of the transition zone has a crucial effect on the velocity distribution, and different transition zone curves lead to the change in the matching relationship between the horizontal velocity of the workpiece and the vertical velocity of the rolls, thus causing the change in the velocity formula. The speed formulas of four research studies of transitional zone curves, such as cubic curve, straight line, and hyperbolic function, are established. Each model has its own characteristics and limitations, but Zhang’s [
42,
43] results are more realistic in that the characteristics and distribution of metal flow velocity in the deformation zone during VGR are described more accurately. Therefore, these results have also been successfully applied to industrial production.
2.2.5. The Rolling Force Formula
At present, there are two types of rolling force models for the VGR process: one is a mathematical model based on the force equilibrium differential equation and the other is a rolling force model based on the energy method.
In references [
30,
39], the contact arc is regarded as a straight line, a common method in cold rolling. In the process of upwards cold rolling, the variation in strip thickness, denoted as 2
y, through the roll gap can be expressed as
y =
ax +
b, where the coefficients a and b can be calculated based on the thickness at the entrance and exit of the roll. Taking into account the influence of the inclination angle
θ, the actual thickness at the exit is expressed as
, Consequently, the value of y at the exit can be given by
.
Given that the deformation is characterized as plane strain and the material is incompressible, we can further obtain
where
represents the horizontal velocity of the element at the x point, while
and
denote the velocity and thickness of the element at the actual exit.
To achieve a straight transitional zone, the equation
must be satisfied, where
vy is the vertical velocity of the rolls. Therefore, Formula (17) can be expressed as [
30]
The general solution to the above equation is
where
C is a constant of integration.
Let
k1 and
k2 be the values of the yield shear stress at the entrance and exit, respectively. At the exit,
,
can be derived based on the actual forward tension stress in the thin area. Thus,
where
. Also, at the entrance,
where
and
qH is the back tension.
Combining Equations (44)–(46) and substituting y = ax + b = H/2 and y = ax + b = h/2 in Equation (44), respectively, produces the following:
For the backward sliding zone [
30],
For the forward sliding zone,
where
,
,
. Through Equations (49) and (50), the distribution of unit pressure along the straight contact arc can be obtained under the condition of the straight transition area at any time during the upwards rolling.
When analyzing downwards rolling, the deformation zone should also be divided into Area I and Area II. The similar solution for Area I during downwards rolling can be written as follows:
For the backward sliding zone [
30],
For the forward sliding zone,
where
,
,
.
,
.
The solution for Area II during downwards rolling is given by
where
. Based on the above equations, we obtained the formula of the roll force as follows:
where
B is the width of the blank,
xn is the coordinate of the neutral plane,
l is the length of the deforming zone, and
xe is the coordinate of the exit in downwards rolling.
Another common treatment for the contact arc in cold rolling is to treat it as a parabola and then solve the force equilibrium equation [
46]. There is no difference between the unit pressure and its subsequent solution, except that the contact arc equation is changed from a straight line to a parabola
y =
ax2 +
b, so that Formula (17) can be expressed as
The general solution to the above equation is
The process of solving the integral constant and determining the boundary conditions to build the rolling force model is consistent with that mentioned above.
The first type of VGR force model is established in the same way as the conventional rolling force model, and there is no innovation in nature. When the force equilibrium differential equation in the deformation zone of VGR is established, different rolling force models are obtained by different contact arc treatments.
Based on the principle of energy conservation, the second type of rolling force model solves the rolling force by calculating the deformation energy, friction energy and other energy changes during the VGR process. Liu [
33] established a mathematical model for roll separating force in the TRB manufacturing process. Based on the minimum energy theory of the variational principle and considering the characteristics of the roll movement and workpiece deformation comprehensively, the internal plastic deformation, friction, shear and tension powers, and the minimum result of the total power functional in UR and DR are obtained. On this basis, Liu [
47] further improved the method above. The elastic deformation of the strip and the flattening deformation of the roll are considered in order to improve the prediction accuracy of the results. The rolling deformation region is divided into elastic deformation and plastic deformation regions, as shown in
Figure 9.
For the elastic deformation region, the roll separating force is calculated using the generalized Hooke’s law. In the plastic deformation region, the roll separating force is calculated using the energy method. By superimposing the roll separating force in the plastic and elastic deformation regions, the total roll separating force values that meet the convergence condition are acquired by using the coupling relationship between the force and the flattening radius.
Based on the above idea, the total rolling force during the upward rolling process is [
47]
where
represents the roll separating force in the entrance elastic region I,
is the roll separating force in the exit elastic region III, and
is the roll separating force in the plastic region II. The specific formulas for these variables are provided in
Appendix A.
Similarly, the total rolling force for the downward rolling process is [
47]
where
represents the roll separating force in the entrance elastic region I during DR,
is the roll separating force in the exit elastic region III during DR, and
is the roll separating force in the plastic region II during DR. The specific formulas for these variables are also provided in
Appendix A.
Three mathematical models and experimental measurements of rolling force were compared, respectively. The results showed that the predictions of the three models are in good agreement with the actual values, as shown in
Figure 10. The experiment used cold-rolled, high-strength, low-alloy-grade steel CR340. The specific material and process parameters are shown in
Table 2.
With the same target workpiece size, TRBs with a final thickness difference ratio of 1:2 were achieved in the above three studies. The rolling force model established using the minimum energy method has better accuracy. The roll separating force calculated by the presentation model is larger than the calculated results using the slab method, and it is slightly larger than the measured data, as shown in
Figure 10d. If the elastic deformation at the entrance and the exit and the flattening of the rolls are also considered on this basis, the accuracy of the obtained model will be further improved. The values calculated by the established model are basically consistent with the measured values, and both sets of data are slightly higher than the results obtained by the engineering method, as shown in
Figure 10b. The rolling force model obtained by solving the conventional force balance equation also has good consistency with the measured values, and the error can be controlled within 10%, as shown in
Figure 10f. In addition, the distribution of unit pressure along the contact arc under the following conditions—UR, DR, and CR—is also compared by using the conventional force balance equation, as shown in
Figure 10e. It can be observed that the unit pressure in conventional rolling falls between that of UR and DR, which is a practical distribution characteristic. This is because the actual exit position of the deformation zone shifts to the left in UR, while it shifts to the right in DR.
Both the traditional method of constructing rolling force models and the use of the minimum energy principle to solve rolling force have been experimentally proven to be reasonable and accurate. Although the first method is based on plane deformation analysis, it is more widely used in engineering practice because it provides a quick means of estimating rolling force. The second method is more suitable for situations requiring high-precision calculations. Obviously, for VGR, the former method may be more appropriate. However, the latter method provides more accurate calculation results, which are also very meaningful and valuable for theoretical research and production guidance.