Novel Analytic Model for the Projected Contact Zone Based on the Flow Line Element Method in Alloyed Bar Rolling

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1. Introduction

Compared with products rolled with the oval-square-oval pass sequence, the bar or rod rolled with round-oval-round pass sequence has a better surface quality and mechanical performance, and the round-oval-round pass sequence is the most common roll pass in bar or rod continuous rolling recently. Since the characteristics of non-uniform distribution of stain, stress and flow velocity on the deformation zone, it is difficult to analyze the process of alloyed bar rolling in oval-square-oval pass sequence accurately [1] [2] [3] .

In the past few years, the research on the alloyed bar rolling process was carried out by the simulation methods and experiments. References [4] [5] [6] [7] [8] studied the strain, stress and geometry of deformation zone of bar rolling by the numerical simulation methods based on the FEM software, and references [9] [10] [11] analyzed the effect of roll gap, roll profile and rolling speed on the wear and exit section area by the single-pass rolling and multi-pass rolling experiments and FE analysis. References [12] [13] [14] studied the rolling force and rolling torque of bar rolling and proposed analytic models for estimating the force energy parameters. Because the contact surface between the rolling workpiece and shaped roll is not a cylindrical surface like flat roll rolling but a three- dimension curve with complex boundary, the length and boundary of contact surface has to be simplified without taking the spread of rolling workpiece into account for obtaining projected area easily. Based on these simplifications, the geometry and projected area of contact zone were calculated by the conventional equation for billet rolling and the roll force and roll torque was derived on these results, and the results of roll force and roll torque were just an approximate value for bar rolling in round-oval-round pass sequence. Furthermore, the profile of contact boundary and the projected area of contact zone influence the results of velocity field distribution, stress field distribution and strain field distribution directly. Therefore, it is indispensable for analyzing the bar or rod continuous rolling process to determine the profile of contact boundary and the projected area of contact zone exactly.

It is not easy to obtain the precise projecting area of contact zone and the accuracy profile of contact boundary since the contour of contact zone is not regular and difficult to be defined. So, at present the geometry and projected area of contact zone is mostly calculated by the simplified methods, such as the empirical equation [2] , the graphical solving method [3] .

Shinokura and Takai [15] [16] [17] gave the spread formula and projected area of the deformed workpiece in round-oval-round pass sequence. However, this formula was derived on the base of rolling experiments of plain carbon steel. Y. Lee proposed a mathematical model for predicting the surface profile of deformed workpiece and mean roll radius in Round-Oval-Round pass sequence [18] [19] [20] .

The profile of contact boundary and the projecting area of contact zone are correlated to the solution of the rolling force, the velocity field distribution, stress field distribution and strain field distribution directly. So, it is very important to determine the profile of contact boundary and the projected area of contact zone exactly.

2. The Existing Models for Determining the Projecting Area of Contact Zone

2.1. Graphical Solving Method [3]

As can be seen in Figure 1, the boundary curve of contact zone was simplified as a intersection line between the revolution surface of roll pass and the profile of incoming workpiece, and the intersection line was obtained by linking the intersect in point one by one. So the projection of contact zone was shown as the hatching zone.

2.2. Analytic Methods Based on the Two Hypothetic Curve Functions for the Contact Boundary

Shinokura and Takai [15] proposed a formula to estimate the maximum contact length L and projected area A_{p} of contact zone in alloyed bar rolling by ignoring the spread of outgoing workpiece.

The size of oval groove, round groove and corresponding incoming workpiece were shown in Figure 2(a) and Figure 2(b) respectively. The equivalent height of incoming workpiece $\stackrel{\xaf}{{H}_{0}}$ and equivalent height of outgoing workpiece $\stackrel{\xaf}{{H}_{m0}}$ were simplified as

$\stackrel{\xaf}{{H}_{0}}=\frac{{A}_{0}-{A}_{s0}}{2{C}_{Z0}}$ (1)

$\stackrel{\xaf}{{H}_{m0}}=\frac{{A}_{0}-{A}_{s0}-{A}_{h}}{2{C}_{Z0}}=\frac{{A}_{e0}}{2{C}_{Z0}}$ (2)

where ${A}_{0}$ is the section area of incoming workpiece, ${A}_{s0}$ , ${A}_{h}$ , ${A}_{e0}$ are the non-effective reduction area, effective reduction area and effective exit section

Figure 1. The projected area by a graphical solving method.

Figure 2. Definition of pass profile and the incoming workpiece. (a) Round-oval pass; (b) Oval-round pass.

area respectively, $\left({C}_{y}{}_{0},{C}_{z0}\right)$ is the intersection point between the profile of incoming workpiece and the roll pass.

In round-oval pass rolling, ${A}_{h}$ and ${A}_{s0}$ may be obtained by

${A}_{h}/2=\mathrm{arctan}\left({C}_{Y0}/{C}_{Z0}\right)\cdot {R}_{a}{}^{2}-\left[{\displaystyle {\int}_{-{C}_{Y0}}^{{C}_{Y0}}\left(\sqrt{{R}_{1}{}^{2}-{y}^{2}}-{D}_{z}\right)\text{d}y}-{C}_{Y0}\cdot {C}_{Z0}\right]$ (3)

${A}_{s0}/2=\mathrm{arctan}\left({C}_{Z0}/{C}_{Y0}\right)\cdot {R}_{a}-{C}_{Y0}\cdot {C}_{Z0}$ . (4)

In oval-round pass rolling, ${A}_{h}$ and ${A}_{s0}$ may be given by

${A}_{h}/2=\frac{\text{\pi}}{8}{W}_{\mathrm{max}}\cdot {H}_{p}-\frac{{A}_{s0}}{2}-\mathrm{arccos}\left({C}_{Z0}/{R}_{Y0}\right)\cdot {R}_{g}^{2}-{C}_{Y0}\cdot {C}_{Z0}$ (5)

${A}_{s0}/2=\mathrm{arcsin}\left({C}_{Z0}/{R}_{1}\right)\cdot {R}_{1}^{2}-\left({D}_{Z}+{C}_{Y0}\right)\cdot {C}_{Z0}$ (6)

where ${R}_{1}$ is the radius of the oval groove, ${R}_{s}$ is the radius of the curvature of the incoming cross-section, ${D}_{Z}$ is the distance along the Z-axis direction between the origin coordinate

Then the maximum contact length ${L}_{\mathrm{max}}$ was obtained by

${L}_{\mathrm{max}}=\sqrt{{R}_{m0}\left(\stackrel{\xaf}{{H}_{0}}-\stackrel{\xaf}{{H}_{m0}}\right)}=\sqrt{{R}_{m0}\frac{{A}_{h}}{2{C}_{Z0}}}$ (7)

where ${R}_{m0}$ is the mean roll radius at the entrance-section, it can be shown as

${R}_{m0}={R}_{\mathrm{min}}+\frac{{H}_{p}}{2}-\frac{\stackrel{\xaf}{{H}_{m0}}}{2}$ . (8)

For obtaining the projected area of contact zone in alloy bar rolling, Shinokura and Takai expressed the contact boundary curve by the function ${L}_{\mathrm{max}}\sqrt{1-\frac{{x}^{2}}{{C}_{y}^{2}}}$ according to the empirical data, and the projected area was shown as

${A}_{p}={\displaystyle {\int}_{0}^{{C}_{y}}{L}_{\mathrm{max}}\sqrt{1-\frac{{x}^{2}}{{C}_{y}^{2}}}\text{d}x=\frac{\text{\pi}}{2}}{L}_{\mathrm{max}}{C}_{y}$ . (9)

As shown in Figure 2, ${C}_{y}$ is the coordinate of critical point on the contact boundary along the spread direction. $\left({C}_{y},{C}_{z}\right)$ is the coordinates of the critical point on the contact boundary at the exit-section, and it can be determined by the reference [16] [17] .

On the base of Equation (9) Y.lee given another contact boundary curve function ${L}_{\mathrm{max}}{\left(\sqrt{1-\frac{x}{{C}_{y}}}\right)}^{1/m}$ to modify this equation and the projected area was shown as

${A}_{p}={\displaystyle {\int}_{0}^{{C}_{y}}{L}_{\mathrm{max}}{\left(1-\frac{x}{{C}_{y}}\right)}^{1/m}\text{d}x=\frac{3}{2}}{L}_{\mathrm{max}}{C}_{y}\left(m=\frac{1}{3}\right)$ . (10)

Moreover, as shown in Equation (9) and Equation (10), two hypothetical functions for indicating the contact boundary curve were given directly without any reasoning and any derivation process. Although the error between the results of two equations are not obvious when the size of rolling workpiece is small enough and then the contact length L and C_{y} is small enough, the absolute error of these two equations will be considerable and it should not be ignored in large diameter bar rolling. So it is not precise enough for these two semi-analytic models to calculate the projected area of multi-pass alloyed bar rolling, then an analytic model should be built to predict the projected area accurately.

In summing up these models for projected area, the Equation (2), which is based on the graphical solving method, does not take the influence of the spread and the contact boundary status of deformed workpiece into account. Therefore, the Graphical solving method can be just used as an estimating value when the spread of deformed workpiece is small enough (Figure 2).

3. A Novel Analytic Model for the Projected Area of Contact Surface

3.1. Modification of Contact Length Model

As shown in Figure 3, if the spread of outgoing workpiece was not negelected, the effective section area ${A}_{e}$ and the equivalent width $2{C}_{y}$ , height $\stackrel{\xaf}{{H}_{m}}$ and mean roll radius ${R}_{m}$ are totally different from the corresponding parameters ${A}_{e0}$ , $2{C}_{y0}$ , $\stackrel{\xaf}{{H}_{m}{}_{0}}$ and ${R}_{m0}$ in Equation (2). $\left({C}_{y},{C}_{z}\right)$ can be calculated by references [19] [20] . So, the contact length modified as

${L}_{m}=\sqrt{{R}_{m}\left(\stackrel{\xaf}{{H}_{0}}-\stackrel{\xaf}{{H}_{m}}\right)}=\sqrt{\left({R}_{\mathrm{min}}+{H}_{p}/2-G/2-\stackrel{\xaf}{{H}_{m}}/2\right)\left(\stackrel{\xaf}{{H}_{0}}-\stackrel{\xaf}{{H}_{m}}\right)}$ . (11)

In round-oval pass rolling, the equivalent height of outgoing workpiece was shown as

$\begin{array}{c}\frac{\stackrel{\xaf}{{H}_{m}}}{2}=\frac{{A}_{e}}{4{C}_{y}}=\frac{{\displaystyle {\int}_{-{C}_{y}}^{{C}_{y}}\left(\sqrt{{R}_{1}^{2}-{y}^{2}}-{D}_{z}\right)\text{d}y}}{{C}_{y}}\\ =\frac{{R}_{1}^{2}\left(2\frac{{C}_{y}}{{R}_{1}}\mathrm{cos}\left(\mathrm{arcsin}\frac{{C}_{y}}{{R}_{1}}\right)+2\mathrm{arcsin}\frac{{C}_{y}}{{R}_{1}}\right)}{4{C}_{y}}-{D}_{z}.\end{array}$ (12)

In oval-round pass rolling, the equivalent height of outgoing workpiece was

Figure 3. Parameters in modified model of contact length. (a) Round-oval pass; (b) Oval-round pass.

expressed as

$\frac{\stackrel{\xaf}{{H}_{m}}}{2}=\frac{{A}_{e}}{4{C}_{y}}=\frac{\mathrm{arctan}\left({C}_{y}/{C}_{z}\right)R{}_{g}{}^{2}+{C}_{y}{C}_{z}}{{C}_{y}}$ . (13)

3.2. Deformation Zone Geometry and the Boundary of Contact Zone

As can be seen in Figure 4 and Figure 5, coordinate axes x, y, z are chosen to be the directions of bar length, width and height, respectively, with the origin of the x-axis at the midpoint of the entry plane. The x-y and x-z are plane of symmetry. The curve equation of contact boundary S can be obtained approximately by interpolating between the point $\left(\text{0,0,}{H}_{0}\right)$ and the point $\left(L,{C}_{y},{C}_{z}\right)$ . The coordinates of any point on the contact boundary S are shown as $\left({x}_{s},{C}_{ys},{C}_{zs}\right)$ .

3.3. The Definition of the Flow Line and Flow Line Element on the Contact Surface

The flow plane was defined as a set of eccentric continuous cylindrical surfaces having almost straight generators parallel to the roll axis. Assuming that at any cross section along the roll bite the bar height deformation is uniform, the deformation zone was constituted by a set of flow plane, and the contact surface was constituted by a set of flow line element ${f}_{\alpha}$ . The flow line element ${f}_{\alpha}$ is a set of concentric circular arc with different radius $R$ and different bite angle ${\theta}_{\alpha s}$ , which center is attached on the roll axis. Since the roll radius in the roll pass is different and the height of incoming workpiece is different along the y-axis direction, the bite angle ${\theta}_{\alpha s}$ , along the whole contact boundary, is not a constant but a variable which changes with a different position angle $\alpha $ or a different roll radius $R$ .

Figure 4. Geometry and flow line element on the contact surface in oval pass. (a) Geometry of contact surface; (b) Flow line on the contact surface.

Figure 5. Geometry and flow line element on the contact surface in round pass. (a) Geometry of contact surface; (b) Flow line on the contact surface.

3.3.1. The Radius of Flow Line R and the Bite Angle θ_{αs} in the Round-Oval Pass Sequence

The radius of flow line ${f}_{\alpha}$ was expressed as

$R={R}_{\mathrm{min}}+{R}_{1}\left(1-\mathrm{cos}\alpha \right)\left(0\le \alpha \le \mathrm{arcsin}\frac{{C}_{y}}{{R}_{1}}\text{or}\text{\hspace{0.17em}}0\le \alpha \le \mathrm{arccos}\frac{{C}_{z}+{D}_{z}}{{R}_{1}}\right)$ . (14)

The three-dimension coordinates of a random point on the contact surface was shown as

$\{\begin{array}{l}x=L-R\mathrm{sin}\theta \left(0\le \theta \le {\theta}_{\alpha s}\right)\\ y={R}_{1}\mathrm{sin}\alpha \\ z=R\left(1-\mathrm{cos}\theta \right)+{R}_{1}\mathrm{cos}\alpha -{D}_{z}=-R\mathrm{cos}\theta +{R}_{c}\end{array}$ . (15)

On the symmetry plane of deformed workpiece, the position angle $\alpha $ is 0 and the bite angle ${\theta}_{\alpha s}$ reaches the maximum value ${\theta}_{\mathrm{max}}$

${\theta}_{\mathrm{max}}=\mathrm{arccos}\left(1-\frac{{H}_{0}-{H}_{p}}{2{R}_{0}}\right)$ . (16)

At the exit section of deformed workpiece, the bite angle ${\theta}_{\alpha s}$ is 0 and the position angle $\alpha $ reaches the maximum value ${\alpha}_{\mathrm{max}}$

${\alpha}_{\mathrm{max}}=\mathrm{arccos}\frac{{C}_{z}+{D}_{z}}{{R}_{1}}$ (17)

According to the Equation (15), the coordinates of points on the contact boundary was expressed as

$\{\begin{array}{l}{x}_{s}=L-R\mathrm{sin}{\theta}_{\alpha s}\\ {y}_{s}={R}_{1}\mathrm{sin}\alpha \\ {z}_{s}=R\left(1-\mathrm{cos}{\theta}_{\alpha s}\right)+{R}_{1}\mathrm{cos}\alpha -{D}_{z}=-R\mathrm{cos}{\theta}_{\alpha s}+{R}_{c}\end{array}$ . (18)

In the contact zone, the height of the profile at $y=0$ along the x-direction is expressed as

$H\left(x\right)={H}_{0}-\frac{2L}{{R}_{0}}x+\frac{1}{{R}_{0}}{x}^{2}$ . (19)

The curve equation of contact boundary S can be obtained approximately by interpolating between the point $\left(0,\text{}0,\text{}{H}_{0}\right)$ and the point $\left(L,\text{}{C}_{y},\text{}{C}_{z}\right)$ . The coordinate of any point on the contact boundary S is shown as $\left(x,\text{}{C}_{ys},\text{}{C}_{zs}\right)$ , and the coordinate ${C}_{zs}$ can be shown as

$2{C}_{z}{}_{s}={H}_{0}-2\left({H}_{0}-2{C}_{z}\right)\frac{{x}_{s}}{L}+\left({H}_{0}-2{C}_{z}\right)\frac{{x}_{s}^{2}}{{L}^{2}}$ . (20)

Substituting the x_{s} of Equation (18) into Equation (20) yields

$2{C}_{zs}=\left({H}_{0}-2{C}_{z}\right)\frac{{R}^{2}{\mathrm{sin}}^{2}{\theta}_{\alpha s}}{{L}^{2}}+2{C}_{z}$ . (21)

According to the equation ${C}_{zs}={z}_{s}$ and ${\mathrm{sin}}^{2}{\theta}_{\alpha s}=1-{\mathrm{cos}}^{2}{\theta}_{\alpha s}$ yields

$\left({H}_{0}-2{C}_{z}\right)\frac{-{R}^{2}{\mathrm{cos}}^{2}{\theta}_{\alpha s}}{{L}^{2}}+2R\mathrm{cos}{\theta}_{\alpha s}+\left({H}_{0}-2{C}_{z}\right)\frac{{R}^{2}}{{L}^{2}}-2\left({R}_{c}-{C}_{z}\right)=0$ . (22)

Solving the Equation (22) yields

${\theta}_{\alpha b}=\mathrm{arccos}\frac{{L}^{2}+L\sqrt{{L}^{2}+\left({H}_{0}-2{C}_{z}\right)\left[\left({H}_{0}-2{C}_{z}\right)\frac{{R}^{2}}{{L}^{2}}-2\left({R}_{c}-{C}_{z}\right)\right]}}{R\left({H}_{0}-2{C}_{z}\right)}$ . (23)

Substituting Equation (14) into Equation (23) yields

${\theta}_{\alpha s}=\mathrm{arccos}\frac{{L}^{2}+L\sqrt{{L}^{2}+\left({H}_{0}-2{C}_{z}\right)\left[\left({H}_{0}-2{C}_{z}\right)\frac{{\left({R}_{\mathrm{min}}+{R}_{p}\left(1-\mathrm{cos}\alpha \right)\right)}^{2}}{{L}^{2}}-2\left({R}_{c}-{C}_{z}\right)\right]}}{\left({R}_{\mathrm{min}}+{R}_{1}\left(1-\mathrm{cos}\alpha \right)\right)\left({H}_{0}-2{C}_{z}\right)}$ . (24)

3.3.2. The Radius of Flow Line R and the Bite Angle θ_{αs} in the Oval-Round Pass Sequence

For the oval-round pass sequence, the radius of flow line ${f}_{\alpha}$ was expressed as

$R={R}_{\mathrm{min}}+{R}_{g}\left(1-\mathrm{cos}\alpha \right)\left(0\le \alpha \le \mathrm{arcsin}\frac{{C}_{y}}{{R}_{g}}\text{or}0\le \alpha \le \mathrm{arccos}\frac{{C}_{z}}{{R}_{g}}\right)$ . (25)

The three-dimensional coordinates of a random point on the contact surface was shown as

$\{\begin{array}{l}x=L-R\mathrm{sin}\theta \left(0\le \theta \le {\theta}_{\alpha s}\right)\\ y={R}_{g}\mathrm{sin}\alpha \\ z=R\left(1-\mathrm{cos}\theta \right)+{R}_{g}\mathrm{cos}\alpha =-R\mathrm{cos}\theta +{R}_{c}\end{array}$ . (26)

On the symmetry plane of deformed workpiece, the position angle $\alpha $ is 0 and the bite angle ${\theta}_{\alpha s}$ reaches the maximum value ${\theta}_{\mathrm{max}}$

${\theta}_{\mathrm{max}}=\mathrm{arccos}\left(1-\frac{{H}_{0}-{H}_{p}}{2{R}_{0}}\right)$ . (27)

At the exit section of deformed workpiece, the bite angle ${\theta}_{\alpha s}$ is 0 and the position angle $\alpha $ reaches the maximum value ${\alpha}_{\mathrm{max}}$

${\alpha}_{\mathrm{max}}=\mathrm{arccos}\frac{{C}_{z}}{{R}_{g}}$ . (28)

By the same methods as round-oval pass sequence, the bite angle of oval- round pass sequence was

${\theta}_{\alpha s}=\mathrm{arccos}\frac{{L}^{2}+L\sqrt{{L}^{2}+\left({H}_{0}-2{C}_{z}\right)\left[\left({H}_{0}-2{C}_{z}\right)\frac{{\left({R}_{\mathrm{min}}+{R}_{g}\left(1-\mathrm{cos}\alpha \right)\right)}^{2}}{{L}^{2}}-2\left({R}_{c}-{C}_{z}\right)\right]}}{\left({R}_{\mathrm{min}}+{R}_{g}\left(1-\mathrm{cos}\alpha \right)\right)\left({H}_{0}-2{C}_{z}\right)}$ . (29)

3.3.3. The Projected Area of Contact Zone

According to definition of projected area of contact zone, it should be calculated by the integral equation as

${A}_{p}=2{\displaystyle {\int}_{0}^{{\alpha}_{\mathrm{max}}}R\mathrm{sin}{\theta}_{\alpha s}\text{d}y}=2{\displaystyle {\int}_{0}^{{\alpha}_{\mathrm{max}}}R\mathrm{sin}{\theta}_{\alpha s}{R}_{1}\mathrm{cos}\alpha \text{d}\alpha}$ . (30)

However, it is too complex for Equation (30) to integrate it directly and get a function for projected area. So, the flow line field method based on the discrete law is an effective way to calculate projected area.

Once the position angle of flow line $\alpha $ is given, the roll radius $R$ and the bite angle ${\theta}_{\alpha s}$ against different position angle of flow line can be determined. Then the space position and the length of any flow line on the contact surface can be obtained. If the whole contact zone was discretized into n flow lines with different position angle ${\alpha}_{i}$ and different bite angle ${\theta}_{\alpha si}$ , the arc length of any flow line was expressed as

${L}_{fi}={R}_{i}{\theta}_{\alpha si}=f\left({\alpha}_{i}\right)\text{}\left(0\le i\le n\right)$ . (31)

The position angle of ith flow line can be shown as

${\alpha}_{i}=\frac{i}{n}{\alpha}_{\mathrm{max}}\text{}\left(0\le i\le n\right)$ . (32)

The projected length of this flow line on plane xoy can be obtained by

${L}_{pi}={R}_{i}\mathrm{sin}{\theta}_{\alpha si}\text{}\left(0\le i\le n\right)$ . (33)

Substituting Equation (32) into Equation (14), Equations ((24), (25) and (29)) to replace the position angle $\alpha $ can yield ${R}_{i}$ and ${\theta}_{\alpha si}$ in oval pass rolling and round pass rolling respectively, then the projected area of contact zone on the plane xoy can be shown as

${A}_{p}=2{\displaystyle {\int}_{0}^{{\alpha}_{\mathrm{max}}}R\mathrm{sin}{\theta}_{\alpha s}\text{d}y}={\displaystyle \underset{i=0}{\overset{n}{\sum}}\left({R}_{i}\mathrm{sin}{\theta}_{\alpha si}\right)}\frac{{C}_{y}}{n}$ . (34)

4. Results and Discussions

The alloyed bar rolling experiments had been accomplished in BEIMAN SPICIAL STEEL CO. LTD, the round workpiece of diameter 171 mm were rolled in one oval pass and one round pass of 22-stand Pomini Rolling Mill. The deformation zone of rolling workpices was obtained by stopping the rolling process when the workpiece was rolled in the oval pass and round pass simultaneously. The rolling schedule is shown as Table 1. The material is structural alloyed steel 41Cr4. The dimension of the groove schedule is respectively shown as Figure 6(a) and Figure 6(b).

As can be seen in Table 2, the parameters for calculating the projecting area of

Table 1. Rolling schedule of the Pomini Rolling Mills.

Table 2. The parameters for calculating the projecting area.

Figure 6. The dimension of the groove schedule. (a) Oval pass (b) Round pass.

Figure 7. Profile of the curve on the contact boundary. (a) Oval-pass; (b) Round-pass.

contact zone were listed one by one.

As shown in Figure 7(a) and Figure 7(b), the profile of curve on the contact boundary in oval pass and round pass were measured and distinguished by a white chalk.

As shown in Figure 8, the contact boundary and the contact surface were obtained by the rigid-plastic FEM software DEFORM5.03. Moreover, the surface profile of incoming workpiece and outgoing workpiece in oval pass and round pass was shown as Figure 9.

Figure 8. Contact boundary and contact zone in oval pass rolling.

Figure 9. Surface profile of incoming workpiece and outgoing workpiece.

As can be seen in Figure 10(a) and Figure 10(b), the 3-dimension contact surface and contact boundary were rebuilt by Matlab 7.0 according to the novel analytic model of contact boundary and contact surface. In Figure 11, the contact boundary from the novel analytic model was compared with that of experimental data and simulation result.

Results of contact length and projected area from the novel model, the calculating results from the existing models, the experimental data and the simulation results were all listed in Table 3. As shown in Table 3, the results of contact length and projected area from different models were listed and compared with corresponding experimental data and simulation results. The results of contact length from the modified analytic model are less than that of empirical formula and the graphical solution, and it is greater than that of Shinokura formula and Y. Lee formula. Moreover, the error of contact length from the modified model is less than that of existing models.

Since the outgoing workpiece of oval pass rolling will be rolled in next round pass as an incoming workpiece and the section profile at the exit of oval pass influence the contact surface of round pass rolling greatly, the prediction error of round pass is obviously greater than the prediction error of oval pass. Moreover,

Figure 10. Rebuilt contact surface based on analytic model by Matlab7.0. (a) Oval-pass; (b) Round-pass.

Figure 11. Profile of the curve on the contact boundary. (a) Oval-pass; (b) Round-pass.

Table 3. Results of different models.

results of projected area from the novel analytic model approaches the experimental data and simulation results very well, and its error is less than any existing models.

5. Conclusions

1) The contact boundary is a complex 3-dimension curve, and its profile is not only concerned with the parameters of pass profile R_{1}, R_{g}, R_{min}, G, D_{z} and the shape parameters of incoming workpiece H_{0}, R_{a}, but also influenced by the coordinates of critical point
$\left({C}_{Y},{C}_{Z}\right)$ ;

2) The modified contact length model is rational because the influence the effective section area of the outgoing workpiece A_{e}, the critical point
$\left({C}_{Y},{C}_{Z}\right)$ on the contact boundary, the effective height of outgoing workpiece
$\stackrel{\xaf}{{H}_{m}}$ and the mean roll radius
${R}_{m}$ , has been taken into account in this model;

3) Based on the different position angle $\alpha $ and bite angle ${\theta}_{\alpha s}$ , the flow line element discretizes the complicated 3-dimension contact surface conveniently and makes it easier to rebuild the contact surface, and it is a good way to analyze the non-uniform stress and strain distribution accurately;

4) The discretizing and summing up method is an efficient way to solve the projected area, and results from this method approach the experimental data and simulating results very well.

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