A Model for Roll-Drawing of Full Sections with Experimental Verification

Roll-drawing of full sections is an alternate process for wire drawing through a closed die, as it conserves less hard tool materials. It therefore enhances the resource efficiency of thin wire production. As the tools are rotating, the relative velocity is lower and friction is reduced. Apart from this, there is only little knowledge about the process mechanics inside the deformation region. According to the relative motion mentioned above, a neutral point should exist within the deformation region, but this hypothesis is unconfirmed until today. In the present work, a combination of an improved empirical model for the lateral spread in roll-drawing and a mechanical model for roll drawing based on the slab method is proposed. The mechanical model deviates from the well-known rolling theory by the fact that a zero deformation torque should exist on the rolls, it predicts the neutral angle, the forward-slip, the external drawing force and the roll separating force. The hypothesis of existence of a neutral point in the deformation region is supported by experimental measurements of the nonzero forward slip in the roll-drawing process. Measurement is done on copper wire, where 200 passes with initial round section sizes ranging from 2 mm to 1.3 mm with an increment of 0.1 mm are evaluated.


Introduction
The roll-drawing process with its characteristic could be a step towards an energy and material efficient production of full-sections, as less hard tool materials and energy is consumed. But compared to the cold rolling process, only few researchers have published knowledge, which can enhance the process planning on industrial level. Especially with respect to the pass design for the roll-drawing process there is a lack of precise spread formulas, but some formulas were proposed in former time [1,2,3]. In the mentioned resources, researchers have observed that the material spread behavior in roll-drawing of carbon steel is quite different to the one observed in hot and cold rolling. For the efficient design of drawing machines and their electrical drive systems, additional process and control models specialized for the roll-drawing process would be helpful. When a pass design for the production of a new full-section has to be developed, one of the first questions is about which initial section has to be taken and additional which force, torque, and power demand can be expected. Often, previously hot rolled round wire rod serves as an initial section for the production of smaller cold drawn full-sections. For the calculation of the groove size and its filling in regular and irregular passes, there in conjunction with an equivalent flat pass method, a precise empirical spread formula is a valuable help to come to a first design. According to that, the passes round to flat and the succeeding flat to flat pass are identified as model passes and investigated in this work. Two new spread formulas are proposed by the authors. For the calculation of force, torque, and power demand of the drives a fast mechanical model for roll-drawing based on the slab method is also proposed in the next sections. For investigation of the lateral spread in roll-drawing, and for the experimental verification of the mechanical model, copper wire (Cu-ETP) is chosen. coefficients of the ordinary differential equation ap , bp , cp , dp parameters of the flowcurve kf flow stress Ax local cross section area kRS scalar influencing factor of free-side surface evolution Ax >0 wire cross section in a distance after the exit plane of the roll gap

Roll gap adjustment
Spread in the second Pass: Flat-Flat. From the set of samples with eight different initial diameters which are deformed by the first pass in the first experiment, see Fig. 4 a) left, the ones with 1.3 mm, 1.6 mm and 2 mm initial diameter are picked out for a further investigation in a second pass. On the samples with deformation from the preceding first round to flat pass, an additional reduction in that second pass is done, see Fig. 4 a) right. Increments of 10 % for the related reduction in height of the first pass and of 5 % for the following second pass are taken. After the first pass, the resulting height, as well as the resulting maximum width, is measured. These values represent the initial sample sizes for the second pass. After the second pass, the final dimensions height and maximum width are measured. In between the first and second pass, the samples are not annealed.
Measurement of roll-drawing force, roll separating force and forward slip: Round-Flat. For the experimental verification of the mechanical model for the calculation of roll-drawing force FT, roll separating force FRS and forward slip κ in roll-drawing measurements are taken. The longitudinally applied roll-drawing force is measured by a load cell clamped between the free end of the wire at the exit of the roll gap and the drawing cart. The roll separating force is measured by load cells placed under the bearing chocks. For the forward-slip measurement, one encoder is attached to the bottom roll for measurement of the roll circumferential speed vcR. The speed of the drawing cart, which is set to be equal to the exit speed of the wire v1, is measured by a second encoder. The roll-drawing is conducted on two different states of the copper wire. One set of data is recorded in an experiment conducted with work hardened copper wire and another set with copper wire which has been recrystallized at a temperature of 450 °C for a time of 30 min at atmosphere.

Measurement of flow stress.
For the verification of the mechanical model, a flow curve of the copper in recrystallized state is needed. For this purpose are taken samples of 6.2 mm in diameter which are recrystallized at a temperature of 450 °C for a time of 30 min at atmosphere. These samples are upset in a compression test to the extent of the logarithmic true strain φ equal to 1.16. For stress and strain calculation the acting force and the travel of the upsetting tool is recorded.

Mathematical Method
Lateral spread calculation in roll-drawing: For calculation of lateral spread in the roll-drawing process only some formulas are published, most of them are having their roots in the hot-flat rolling. The spread formula of Hill [5] is given as Eq. 1. Jaschke [1] is using the formula of Hill with the coefficient 0.525 instead of 0.5 which has been proposed by Limant [4], additionally he is using the final width b1 on the right-hand side of the equation instead of initial width b0, see Eq. 2. The length under direct pressure ld is calculated with Eq. 3: . ( . . (3) Lambiase [3] proposed for the prediction of the degree of spread β in roll-drawing Eq. 4 for steel samples he investigated, where the initial wire diameters d0 were 4.6 mm and 5.3 mm and the diameter of the idle rolls dR 80 mm.

Achievements and Trends in Material Forming
.
For the prediction of spread in roll-drawing of copper wire, two new equations are proposed by the authors. For the first pass, forming the round section to the flat shape with curved side surface Eq. 5 and for the second pass forming the flat into a flat Eq. 6 is proposed. . .
Calculation of roll-drawing force, roll separating force and forward slip: Round-Flat pass. For the prediction of the necessary force, torque and power demand of a drawing machine used for the rolldrawing process a mechanical model based on the slab method is given in the following section, for flat wire production and flat rolling, such type of models are known [6,7]. The main geometrical dependencies in the roll gap are derived with the help of Fig. 2. The acting stresses and geometric properties of the infinitesimal slab element, which are relevant to form the equilibrium of forces, are shown in Fig. 3. In Fig. 4 b) the geometric properties for the model of the local cross-section area Ax of the wire in the gap are shown. Considered is the symmetric case with respect to friction coefficient μ as well as the idle roll radius of top RRt and bottom roll RRb.
The equations Eq. 7 to Eq. 11 describe the geometrical relations in the roll gap. With the position dependent height hx from Eq. 10 the position dependent maximum true strain φ in height direction is calculated, given in Eq. 12. . . . . . .
(12)   Geometry of local area Ax and wire dimensions. With Eq. 13 a local representation of the length under direct pressure ldx is calculated, which is necessary for the calculation of the local maximum width bmax along the roll gap coordinate x, represented by Eq. 14. The local distribution or evolution of the shape of the free-side surface along the roll gap is modeled with Eq. 15. The scalar value kRS is dependent on the reduction in the pass. The evaluation of the kRS value is described in a later section. For the calculation of the position dependent width of contact area bmin x from the position dependent maximum width bx and with the help of the shape of the free-side surface, described by the radius of the free-side surface RSx, the inner length aS and the outer length dS of the circular section have to be calculated in advance. The local area Ax at each position in the roll gap is subdivided into a center part of rectangular shape and two circular sections ACS on the sides, see Eq. 20. . . . . . . .

Achievements and Trends in Material Forming
. (20) The force equilibrium on the slab in x-direction is represented by Eq. 21, which is derived from the acting stresses given in Fig. 3. After canceling out the width b and a bit of rearrangement one comes to Eq. 22. For the following formulas, a general friction coefficient ̅ μ is defined, where the positive sign has to be used in the forward-slip zone and the negative sign in the backward-slip zone, see Eq. 23. With Coulomb's friction law, Eq. 24, and the force equilibrium on the slab in y-direction, Eq. 25, the horizontal stress σx and the vertical stress σy are linked. For the description of the normal stress in y-direction as a function of the longitudinal normal stress, Tresca's yield criterion is used in the given way, see Eq. 27. Following some further rearrangement, Eq. 28 is formed. By placing the appropriate signs in front of the friction coefficient, for forward and backward-slip zone, in Eq. 29 and Eq. 30 one comes to the two differential equations, one valid for the region in front of the neutral point and one for the region behind.
. (21) . ( . . . . . . . . (30) The solution of the differential equation in x-direction is leading to the longitudinal stress distribution in the roll-gap, σx (x), from which with the help of Eq. 27 and Eq. 43 the vertical stress distribution σy (x) in the gap can be calculated. By integration of the product of the vertical stress distribution σy (x) and a mean value of the width of contact area bmin x and the maximum local width bx along the gap coordinate x the roll separating force FRS is calculated, see Eq. 31. For the resulting rolling torque in the roll gap TR the quite similar formula, Eq. 32, is derived. In Fig. 5 the different torques acting on the roll are shown. There is the rolling torque TR, the bearing torque TB as a result of the friction acting in the bearing, the external torque TT coming from the application of the drawing force which can be described also by a stress acting on the cross-section area of the wire in the exit plane of the roll gap and an additional virtual torque from an external drive TD. These four torques have to be in equilibrium, which is represented by Eq. 35. For the roll-drawing case, the external torque from a drive TD has to become to zero. . . .
The following equations Eq. 36 and Eq. 37 are describing the longitudinal component of the roll circumferential speed vxR and the longitudinal speed of the drawn wire vx at each position in the roll gap. . .
The forward-slip is defined by Eq. 38. A stretching of the material after the roll gap can be identified by the value of the degree of stretch λext deviating from one. The external plastic stretch λext is calculated with Eq. 39, where the measured drawing velocity v1ms is related to the calculated velocity vx=0 of the wire at the exit of the gap. The influence of such an external plastic stretch on the cross-section of the wire Ax >0 measured in some distance to the exit plane of the roll gap is calculated with Eq. 40. The influence of such an external stretch on the wire dimensions, maximum width bx >0

Achievements and Trends in Material Forming
and minimum width bmin(x >0), measured in some distance to the exit plane of the gap is considered by Eq. 41 and Eq. 42. . . . . .

(42)
Flow curve evaluation for the verification of the mechanical model. On basis of a least squares method, the parameters of the given model function for the cold flow curve, Eq. 43, of the copper wire used in the experiments are evaluated. The true stress kf is here only a function of the logarithmic true strain φ. . (43)

Results and Discussion
Lateral spread calculation in roll-drawing: For the prediction of the lateral spread of copper flat wire in a first pass where an initial round is formed into a flat with curved side-surface, the above proposed Eq. 5 is used, for the investigated range of initial diameters. Eight sets of five free model parameters A to E have been calculated by a least squares method. The parameters belonging to the different initial diameters d0 as well as the resulting R 2 values can be seen in Table 1. As examples for the correlation between predicted and measured width values the diagrams in Fig. 6 and Fig. 7 are given, shown are the results for 1.6 mm and 2.0 mm initial wire diameter. The behavior of the copper wire in roll-drawing is in general comparable to what Lambiase [3] has observed with carbon steel wire samples having bigger initial dimensions. Some of the parameters A to E, like for example parameter D is nearly constant for the whole range of initial wire diameters d0 whereas others, like parameter B is strongly fluctuating. But with respect to the overall good R 2 values, one can rely on the calculated final width values. A small drawback is the fact that for the calculation of the final width b1 with Eq. 5 a numerical solution method has to be used.   For the prediction of lateral spread in roll-drawing where a preformed flat is further deformed in a second pass in a flat to flat fashion, the above proposed Eq. 6 can be taken. The results for the seven model parameters A to G, to be used in conjunction with Eq. 6 and the resulting R 2 values are shown in Table 2. As examples for the quality of the prediction, the plots for the sample sizes with initial height of 1.81 mm and initial width of 2.04 mm out of the 2 mm initial diameter set of samples and the results for the sample with the initial height of 0.95 mm and initial width of 1.95 mm from the 1.6 mm initial diameter set of samples are shown in Fig. 8 and Fig. 9.

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Achievements and Trends in Material Forming Table 2: Parameters for the prediction of the degree of spread in the flat to flat pass with the proposed Eq. 6. Fig. 8: Predicted degree of spread and measured degree of spread for the sample with 1.81 mm initial height, R2 = 0.996. Fig. 9: Predicted degree of spread and measured degree of spread for the sample with 0.95 mm initial height, R2 = 0.878. The accuracy of the prediction of lateral spread with Eq. 6 for the flat to flat roll-drawing pass of copper wire is quite high. Even the worst parameter set, with respect to the R 2 value, is capable to describe the general spread behavior. Eq. 6 has, compared to Eq. 5 the benefit, that it can be used for fast calculation on a simple pocket calculator.

Calculation of roll-drawing force, roll separating force and forward slip: Round-Flat pass.
For the prediction of roll-drawing force FT, roll separating force FRS and the forward-slip κ in rolldrawing with the proposed model and its validation roll-drawing experiments on recrystallized copper are conducted. The recrystallized copper is chosen to have a known initial state with respect to the present flow stress kf of the material. The strain dependent flow stress is evaluated from a compression test. In Fig. 10 the flow stress kf approximated by an exponential function is plotted over the logarithmic true strain φ. Results from conducted roll-drawing passes are shown in the following section. As the initial section a 2 mm round wire serves, which is deformed to four different heights h1, always starting from the initial round section. The measured final dimensions are given in Table 3. The graphical representations of the results, for the different parts of the proposed model are exemplarily given for the second reduction in height.  In the model, the knowledge about the local area Ax is crucial. It is composed of the rectangular center part and the two curved side parts, remember Fig. 4. At first the local maximum width bx is calculated, the result is shown in Fig. 11. For the calculation of the area of the curved side parts and for the calculation of the local minimum width bmin x the side radius RS has to be known, which is modeled by Eq. 15. In roll pass design practice, for calculation of groove filling in hot rolling operations, it is often the case that this side radius is set to be constant and a parallel shift along with the maximum width is performed which leads to good results. Here it would lead to quite big error on the resulting local area Ax, which can be seen from Fig. 12, where the local area is shown as a function of the local free side-surface radius RSx. The uppermost line shows the case where the side radius RS is set equal to the initial radius of the wire RS0 and kept constant over the whole length of the roll gap. It can be Table 3: Initial height h0, initial width b0, final height h1 and final width b1 of four passes.

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Achievements and Trends in Material Forming seen that this would lead to an increase of area even with respect to the cross-section at the exit of the gap. A question is how the change of the side radius along the contact length happens, in other rolling experiments conducted by the authors, on bigger sample sizes, it has been observed that the change of the side radius RSx is non-linear along the gap coordinate x but shows a quasi linear dependency with respect to the local absolute change in width Δbx . That is assumed to be also the case for the roll-drawing. To which extent that change of the shape of the free side-surface is acting, can be described by the scalar influencing factor kRS, see Eq. 15. As an indication for a reasonable choice of the value of influencing factor kRS the plot of the resulting minimum width bmin on the exit side of the gap can be used, see Fig. 13. For the model validation measurements of the minimum width bmin at the final sections are done by taking close up photograph of the wire top surface and relating the counted number of pixels to real world dimensions. The applied values for kRS and the values of the resulting minimum width bmin from the model calculation at the exit plane of the roll gap are collected in Table 4.   The local side radius RSx of the free side-surface calculated with Eq. 15 is plotted and shown in Fig. 14, beginning with the initial side radius RS0 equal to 1 mm and decreasing values along the gap Table 4: Applied kRS values and resulting minimum width values bmin at the exit plane of the roll gap of the treated four passes. coordinate x, like it has been observed in the rolling case, the side surface is sharpened. The local minimum width bmin along with the gap coordinate x is shown in Fig. 15, it starts from the zero value at the inlet plane and rises to its maximum value, which is reached at the exit plane of the roll gap.
With the values of the local minimum width bmin x and the local side radius RSx together with the height hx along the gap coordinate x the local area Ax can be calculated, the result is shown in Fig. 16.   In the next step, the acting stress distributions in the roll gap are calculated. For that purpose, at first the flow stress kf of the material described by Eq. 43 is evaluated with the locally dependent true strain φ calculated from Eq. 12. The normal stress in longitudinal direction σx (x) is found by solving point by point the differential equation, Eq. 28, on a dense discretized point space with a numerical ODE solver. When the stress distribution σx (x) in the forward-and backward-slip zone is found, the neutral point can be found by bringing the two solutions to intersection, see Fig. 17. By using Tresca's yield criterion, given by Eq. 27, also the normal stress distribution σy (x) can be evaluated. The shear stress distribution τx (x) is calculated with help of Eq. 24 and Eq. 26. The friction coefficient μR acting between roll and wire is adjusted to the values shown in Table 5. In the combined plot shown in Fig. 18, all stresses acting along the gap can be seen.

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Achievements and Trends in Material Forming With the normal stress σy and the local maximum and minimum width bx and bmin x the roll separating force FRS is calculated from Eq. 31. By using the shear stress τx and the local widths, the rolling torque TR is calculated from Eq. 32. With the help of the roll separating force FRS and the acting friction coefficient μB in the bearing, the bearing torque TB is calculated. From Fig. 5 the equilibrium of the acting torques can be seen. For the roll-drawing case, the shown torque of an external drive TD has to become to zero. That case is tracked by Eq. 35. With the measured drawing force FTmeas the applied stress σx on the outlet of the gap has been calculated, which gave the initial condition for the stress distribution in the forward-slip zone, after that the friction coefficient μR has been adjusted in a way that the calculated roll separating force came as close as possible to the measured value, while the condition for having the value for the virtual torque of an external drive TD in the ideal case exact equal to zero is more important and maintained by adjusting the friction coefficient μB acting in the bearing. The resulting values of the calculation and the measured ones are collected in Table 5. Table 5: Measured and calculated forces, torques, and friction values at the contact between drawn material and rolls and in the journal bearing of the four roll-drawing passes.
With the intersection point between forward-and backward-slip zone one finds the neutral point position xN. Here, at the point of no slip, the longitudinal velocity vx of the drawn material and the longitudinal component vxR of the roll circumferential speed vcR are equal, now also the neutral angle αN can be calculated. With Eq. 36 and Eq. 37 the velocities of the wire and the roll with respect to the roll gap coordinate x are calculated. Their relation to each other can be seen from Fig. 19. The calculated maximum of the velocity difference is present at the exit plane of the roll gap, which was expected, but the absolute value of the calculated forward-slip κ is to small compared to the measured values. The definition of the forward-slip κ is given by Eq. 38 and its graphical representation along with the roll gap coordinate x is shown in Fig. 20. The difference between the measured forward-slip and the calculated forward-slip could be explained with a free homogeneous stretch acting on the material close to the exit plane of the roll gap, which influences also the final maximum width bmax to some extent. That velocity difference or mismatch is considered within Eq. 39. It is captured by an external plastic stretch λext the wire additionally undergoes after leaving the roll gap. The measured and the calculated forward-slip values and the external stretch λext is shown in Table 6. With Eq. 40 the influence of that external plastic stretch onto the area of the wire which is present exactly at the exit plane of the gap is described. The area reduced by that acting plastic stretch is then taken for the calculation of a new reduced maximum width bx>0 behind the gap, for that Eq. 41 is taken. The influence on the minimum width bmin is treated with Eq. 42. The results of the comparison of calculated and measured values of the maximum width bmax are given in Table 7. The calculated minimum width bmin-c at the end of the gap, the minimum width calculated and the minimum width in a distance to the exit plane of the gap bmin x>0-c and the minimum width measured within the close up photograph of the samples bmin-ms and a comparison is given in Table 8. Table 6: Measured forward-slip κms and calculated forward-slip κc and the calculated external plastic stretch λext acting after the roll gap.

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Achievements and Trends in Material Forming Table 7: Measured maximum width b1-ms and calculated maximum width b1-c at the end of the gap and after considering the external stretch λext. Table 8: Measured minimum width bmin-ms and calculated minimum width bmin-c at the end of the gap and after considering the external stretch λext.
The deviation between the measured maximum width b1-ms and the calculated maximum width b1c at the exit plane of the gap shows a mean positive deviation of 1.2 % for the four investigated passes. After the consideration of the external plastic stretch λext a quite bigger mean negative deviation of 1.27 % is found. For the minimum widths the deviations at the exit plane are all positive at a mean value of 6.94 %. The consideration of the plastic external stretch reduces the absolute mean deviation to 1.13 %. The influence of that external stretch onto the radius of the free side-surface is neglected in Eq. 41 and Eq. 42.

Conclusion and Outlook
The proposed model functions for the prediction of lateral spread in the investigated roll-drawing passes, initial round section to flat with curved side-surface and flat to flat, performs good in the examined dimension range. The formulas are not too complicated to handle and can enhance the roll pass design, especially they could be helpful for the choice of the appropriate initial wire dimension for a desired final dimension to be manufactured by the roll-drawing process. Additionally, they could be used for the rough calculation of the groove filling behavior in profiled wire drawing. The mechanical model for the prediction of roll separating force, external roll-drawing force and neutral angle shows promising results. The forces and torques are strongly dependent on the assumed friction coefficients and the minimum width, which are values not known in advance. With the proposed model equations in conjunction with the numerical values given for the prediction of the curvature radius of the free-side surface an educated guess should be possible of the roll pass designer. To come to a reliable calculation scheme over a bigger size range, more tests as well as parameter studies have to be conducted. The local area described by local maximum width, local minimum width and the position dependent curvature radius of the free side-surface should be investigated to a deeper extent. The next experiments will have also to take the high forward-slip further into focus, which lead to the proposed action of an external stretch, acting close behind the exit plane of the roll gap, and showing a remarkable influence onto the final wire geometry.