Analysis of High-Speed Rolling With Inertia and Rate Effects

Limit-analysis procedures for time-dependent materials are utilized for assessing some essential technological parameters in high-speed strip rolling (i.e., the torque, the separation force, the minimal friction required to avoid skidding, maximum allowable speed, etc.). The formulations are quite wide in scope (e.g. they include the inertia of the plastic flow beside the material rate effect) but lack, in general, the rigor of the true bound by reasons to be discussed. The solutions are, by default, considered as “approximate bounds” unless stated differently. Due emphasis is given to the development of a lower bound, infrequently employed in metalworking analysis. It yields relevant information about the process which appears entirely consistent with an independent upper bound solution. In particular, the rate effects are shown (in both solutions) to be characterized by the intensity of two dimensionless groups, known universally as Bingham No. and Euler No. Normally they cannot be ignored at high speeds currently attainable in modern industry, above, say, u₀ = 50 [m/s]. For slow speeds, the above solutions constitute rigorous upper and lower bounds. The relative close proximity of the two bounds to experimental data (with copper, aluminum, and steel) and their excellent agreement with the rigid-plastic finite element solution, demonstrate the utility of having these dual bounds simultaneously. A seemingly useful by-product from the analysis is the ability to predict the onset of skidding at very high speeds. For this sake, an expression is offered for determining the maximum allowable rolling speeds (at the incipient of skidding) in conjunction with the requirement for a certain minimum interfacial friction.