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In this work, slab method and upper bound method are proposed to investigate the twist compression forming of cylinder. Both upper bound method and slab method assume the friction between the cylinder and the die as constant shear friction (τ = mk). The total power rate of the energy consumption with a 3D velocity field based on the upper bound method is constructed. An optimum variable parameter is obtained by using the energy minimization method. The upper and lower dies are rotated in different directions, so the cylinder is compressed with the twist circumstance, and the frictional shear stresses (τ) are not acted in the radial direction. The stresses (τ) are acted in an arbitrary direction at angle α from the radial direction relative to the r-axis. According to the velocity relations between radial direction and circumferential direction, angle ( can be determined. Combining the upper bound method with the slab method, the effects of frictional factor, aspect ratio, height reduction, and angular velocity etc on the twist compression forming characteristics are explored effectively. The feasibility of both models can be realized by comparisons of two models and the experiments.

An analytical modified model of the cold bonding rolling of unbounded double-layers sheet is formulated to obtain the stress field of the double-layers sheet at the roll gap via the direct derivation without using Runge–Kutta numerical method. Generally Coulomb friction at the interface of unbounded sheets and between rolls and double-layers sheet are assumed. However, when the frictional coefficient at the interface of unbounded sheets is high enough, the friction needs to be modified. Thus hybrid friction should be taken into account at the interface so as to obtain the realistic rolling characteristics. In this study, the neutral point between the rolls and the double-layers sheet, the rolling pressure distribution along the contact interface between the rolls and the double-layers sheet, the horizontal stress of the whole double-layers sheet, the horizontal stresses in the component layers, the realistic shear stress at the interface of double-layers sheet, the rolling force, and the rolling torque, etc., can be calculated easily and effectively. Besides, obtaining the bonding point at the interface and the thickness ratio of the clad sheet at the exit are of great important. Moreover, the bonding conditions of the cold bond rolling of the unbounded double-layers sheet can be found to provide useful knowledge in carrying out the bond rolling to prevent the failure. In summary the complete formulae of the rolling pressure distribution, the rolling force, the rolling torque can be derived by this analytical modified model. This analytical approach proposed is suitable for the on-line bond rolling.

The study mainly aims at the asymmetrical PV 1 cold rolling of sheet to investigate the minimum thickness. For the purpose of deriving the minimum thickness, the complete equation for asymmetrical cold rolling of sheet is derived by an analytical approach based on the slab method. Two analytical models for the minimum thickness of asymmetrical PV cold rolling are established, Model I considering the same frictional coefficients for the upper and lower rolls, whilst Model II considers different frictional coefficients for the upper and lower rolls. The relationships between the ratio of effective radius to initial thickness and bite angle for the two models are proposed. The comparisons of front tension, rolling force per unit width, reduction, and limiting roll speed ratio for the two models are examined. From an analytical approach, the values of minimum thickness can be obtained. The study proposes a new analytical technique to explore the minimum thickness.