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The authors have declared that no competing interests exist.

Conceived and designed the experiments: HY XL. Performed the experiments: DT MS. Analyzed the data: HY XL DT MS. Contributed reagents/materials/analysis tools: XL. Contributed to the writing of the manuscript: DT HY.

Parts produced by microforming are becoming ever smaller. Similarly, the foils required in micro-machines are becoming ever thinner. The asymmetric rolling technique is capable of producing foils that are thinner than those produced by the conventional rolling technique. The difference between asymmetric rolling and conventional rolling is the ‘cross-shear’ zone. However, the influence of the cross-shear zone on the minimum achievable foil thickness during asymmetric rolling is still uncertain. In this paper, we report experiments designed to understand this critical influencing factor on the minimum achievable thickness in asymmetric rolling. Results showed that the minimum achievable thickness of rolled foils produced by asymmetric rolling with a rolling speed ratio of 1.3 can be reduced to about 30% of that possible by conventional rolling technique. Furthermore, the minimum achievable thickness during asymmetric rolling could be correlated to the cross-shear ratio, which, in turn, could be related to the rolling speed ratio. From the experimental results, a formula to calculate the minimum achievable thickness was established, considering the parameters cross-shear ratio, friction coefficient, work roll radius, etc. in asymmetric rolling.

Micro-manufacturing has attracted increasing attention over recent years due to consumer-driven and industry-driven trends towards product miniaturisation in applications such as engineering and medicine

Asymmetric rolling is a severe plastic deformation technique that can refine grain size in sheet/foil materials. This technique has been used to improve the mechanical properties of products

The asymmetric rolling technique can produce thinner foils. In conventional rolling, the foils produced cannot be thinner than a certain value. When this happens, there is no method to reduce the foil thickness further. Even increasing the rolling force or the number of rolling passes does not help. In addition, the quality of products can suffer especially when the rolling mill has been operating close to its full capacity for a long time. Therefore, foil rolling becomes a technical problem when the foil is required to be thinner. This thickness limit is called the minimum achievable thickness. Two factors can influence the minimum achievable thickness: (i) _{min} is minimum achievable thickness; _{0}

The deformation brought about by asymmetric rolling is different from that in the case of conventional rolling - Asymmetric rolling involves formation of the cross-shear zone. As seen above, some research on the mechanical properties of foils during asymmetric rolling has been carried out. However, there has been no reported research on the minimum achievable thickness of foils during asymmetric rolling. In this paper, we describe a novel method to measure and analyze the minimum achievable thickness during asymmetric rolling. The experiments were conducted using a four-high asymmetric roll mill. The effect of rolling parameters such as rolling speed ratio and cross-shear ratio on the minimum achievable thickness was analyzed in these experiments. Finally, we propose a novel theoretical model on the basis of the experimental results. The results provide a mathematical foundation for further studies of the minimum achievable thickness in asymmetric rolling.

In the experiments, a four-high experimental asymmetric roll mill was employed. The main parameters of the mill are listed in

Mill type | Work rolling diameter [mm] | Work rolling length [mm] | Backup rolling diameter [mm] | Backup rolling length [mm] | Maximum rolling force [kN] |

Ф50 Four-high asymmetric mill | 50 | 130 | 120 | 120 | 200 |

(1. Screw-down device; 2.Backup roll; 3.Work roll; 4.mill house; 5.Tensile motor; 6.Universal shaft; 7. Main drive motor).

The roll mill described above was used in the experiments. The employed coils were made of Q195 steel, whose composition is listed in

Q195 | C | Mn | Si | S | P | Fe |

w% | ≤0.12 | ≤0.50 | ≤0.30 | ≤0.040 | ≤0.035 | Balance |

In the experiments, multi-pass rolling was carried out until the foils were rolled to the minimum achievable thickness. In order to avoid the influence of other parameters on the minimum achievable thickness, only the rolling speed ratio was changed, with the materials, friction condition, work rolls kept fixed. Samples with initial thickness 0.35 mm were rolled to the minimum achievable thickness with rolling speed ratios of 1.0, 1.1, 1.2 and 1.3. All rolling experiments were without application of tension. The experiments would not be stopped until the thicknesses of foils were unchanged for the last three passes, for all four rolling speed ratios. After each pass, the thickness of rolled sheets was measured with a micrometer at 3 different points. The average thickness value after each pass was recorded, in addition to the linear velocities of work rolls, exit thickness of foil and exit velocity in each rolling pass. The roll speed ratio was a vital parameter for adjusting the cross-shear ratio.

The cross-shear ratio is the ratio of the area of the cross-shear zone to the area of the whole plastic deformation region, as shown in _{1}_{2}_{1}>V_{2}_{1}>>V_{2}

We used the ‘nick’ method to measure the entrance and exit speeds of the sheet, as shown in _{i}_{1}), work roll radius (_{entry}_{exit}

In order to calculate the cross-shear ratio, it is important to calculate the angles _{1}, γ_{2}

As shown in

The influence of elastic deformation of work rolls on the biting angle in the rolling deformation zone is ignored.

In order to calculate _{1} and _{2}, we make the following assumptions:

1) The friction coefficients between the strip and the work rolls are same and constant; and the rolling pressure is uniform in the rolling deformation zone

2) The cumulative stress along the rolling direction is zero, so that:

In the rolling deformation zone,

Thus, _{x_b}_{x_f}_{x}_{x}

3) Due to the very small thickness of foils, here we assumed that the mean rolling velocity nearly equals that at the neural surfaces, according to the principle of equal flow at two neutral surfaces. Thus, assuming that the mean rolling speed at neural surface of _{1} equals the surface speed of the upper roll, and the mean rolling speed at neural surface of _{2} equals the surface speed of lower roll. Thus, referring to

If there only the backward slip zone and the cross-shear zone exist in the rolling deformation zone, Eq. (9) can be simplified as:

If there are only the cross-shear zone and the forward slip zone in the rolling deformation zone, Eq. (9) can be simplified as:

Lastly, if only the cross-shear zone exists in the rolling deformation zone, Eq. (9) reduces to:

In the experiment, the _{1}, _{2}, _{entry}_{exit-up}_{exit-down}_{1} and _{2} could be obtained. The cross-shear ratio could be obtained from Eqs (4) and (5).

Rolling pass | 1.0 | 1.1 | 1.2 | 1.3 | ||||

Thickness | Error | Thickness | Error | Thickness | Error | Thickness | Error | |

10 | 54.0 | 0.8 | 46.7 | 1.2 | 25.0 | 1.6 | 18.3 | 0.4 |

11 | 53.7 | 1.2 | 45.7 | 0.4 | 24.7 | 2.0 | 18.0 | 0.0 |

12 | 53.7 | 0.4 | 45.7 | 0.4 | 24.7 | 1.2 | 18.0 | 0.8 |

As described before, in asymmetric rolling, the cross-shear ratio was a vital parameter related to the rolling speed ratio. Three groups of parameters such as entrance and exit velocities, entrance and exit thicknesses and the roll velocities were recorded from the 4th pass to the 11th pass. These were used to calculate the cross-shear ratio in the rolling process.

Rolling pass | Slow roll velocity [mm/s] | Fast roll velocity [mm/s] | Exit thickness [µm] | Entry velocity [mm/s] | _{exit-up} |
_{exit-down} |
Cross-shear ratio |

0 | 39.4 | 43.4 | 343.6 | - | - | - | |

1 | 38.7 | 42.9 | 284.6 | 32.0 | 39.3 | 43.5 | 0.159 |

2 | 38.6 | 42.9 | 220.3 | 30.9 | 39.3 | 46.2 | 0.207 |

3 | 38.4 | 42.9 | 159.3 | 32.6 | 39.4 | 45.4 | 0.227 |

4 | 38.4 | 43.0 | 122.6 | 41.2 | 47.8 | 55.5 | 0.285 |

5 | 38.4 | 43.0 | 98.0 | 34.2 | 38.7 | 44.9 | 0.287 |

6 | 38.4 | 42.9 | 80.0 | 36.0 | 39.0 | 44.6 | 0.288 |

7 | 38.4 | 42.9 | 69.0 | 36.4 | 38.7 | 44.9 | 0.462 |

8 | 38.3 | 42.9 | 60.0 | 36.9 | 38.7 | 44.3 | 0.592 |

9 | 38.3 | 43.2 | 53.3 | 36.4 | 38.4 | 44.9 | 0.651 |

10 | 38.2 | 43.3 | 46.6 | 40.9 | 38.5 | 45.0 | 0.695 |

11 | 38.2 | 43.3 | 45.6 | 41.9 | 38.5 | 45.3 | 0.695 |

12 | 38.2 | 43.2 | 45.6 | 41.5 | 38.5 | 45.3 | 0.695 |

Rolling speed ratio | 1.0 | 1.1 | 1.2 | 1.3 |

Cross-shear ratio for 11th rolling pass ( |
0 | 0.695 | 0.810 | 0.94 |

In conventional rolling, the minimum achievable thickness is approximately proportional to the diameter of the work rolls and the deformation resistance of the sample material. However, in asymmetric rolling process, the minimum achievable thickness is also affected by the rolling speed ratio and the cross-shear ratio. It is seen in

In asymmetric rolling, the difference of linear velocities of the two work rolls leads to the plastic deformation region being divided into three parts: (1) the forward-slip zone, (2) the cross-shear zone and (3) the backward-slip zone, as shown in

To simplify the formulation involved in the analysis, the following assumptions and simplifications are made:

(1) The rolling process is approximately one of flat compression;

(2) The plastic deformation is a state of plane strain;

(3) Friction forces on the contact surfaces are given by Coulomb's law of friction.

(4) The tension forces at the entrance and exit are equal in magnitude, which ensures that the length of the forward-slip zone is equal to the length of backward-slip zone.

As shown in _{1}+l_{2}_{2}_{1}

According to the Stone formula

The unit average rolling force in the cross-shear zone:

The unit average rolling force in the backward-slip zone:

Note that only the value of the cross-shear ratio was considered. So it is assumed that _{1} = l_{2} = l/4

Considering elastic deformation of the foil and rolls, the Hitchcock equation

Let

and

Then,_{f}_{c}_{b}_{1} and _{2} Young's modulus of workpiece and work roll material respectively; Δ_{1} and _{2} are Poisson ratio of workpiece and work roll materials respectively.

If Eq. (18) has a positive root, and the _{c}

When _{c}_{c}_{c}

In _{c}_{c}

Eq. (19) has a unique solution when _{c}_{c}

In asymmetric rolling, _{c}^{+} to 1.5441 as ^{+} to 1. When _{c}

From the geometry, _{1} can be calculated because _{1}, _{2} and _{exit}

In

Thus, the minimum achievable thickness by asymmetric rolling could be transferred into Eq. (24) from Eq. (20).

Eq. (24) shows that the minimum achievable thickness of foil in asymmetric rolling overcomes the limitations imposed in conventional rolling, due to the creation of the cross-shear zone. The minimum thickness of foil in asymmetric rolling is a function of the cross-shear ratio, the coefficient of friction between the foil and the rolls, the diameters of the work rolls and the deformation resistance of the foil. As the rolling speed ratio increases, the cross-shear zone increases in size, leading to greater foil reduction during the pass, and consequently the minimum thickness decreases.

Eq. (24) can predict the minimum achievable foil thickness during asymmetric rolling. Based on the experiment, _{0} = 2.3×10^{−11}; the yield strengths estimated by tensile tests are 230 MPa, 240 MPa, 249.6 MPa and 260.9 MPa for the last exit thicknesses for rolling speed ratios 1.0, 1.1, 1.2 and 1.3, thus the values of

Experimental results show that the minimum achievable thickness achievable by asymmetric rolling with the rolling speed ratio 1.3 is 30% of that possible by conventional rolling.

A new formula, Eq. (24), has been developed to predict the minimum achievable thickness (_{min}

The cross-shear ratio is related to the rolling speed ratio, the entry and exit speeds of the foil and the linear speed of the upper and lower rolls. When the deformation region is made up of three parts, the cross-shear ratio can be calculated as shown in Eq. (22). As the foil thickness decreases, the exit speed of the foil tends to the linear speed of the slower work roll and the cross-shear ratio increases in multi-pass asymmetric rolling.

YU H.L. thanks Dr. Ajit GODBOLE of the University of Wollongong for discussing the paper.