An Analytical Rolling Model Including Through Thickness Shear Stress Distributions

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Abstract
An extension of the slab method of analysis as applied to rolling is presented. The von Mises yield criterion, including shear stresses, is written in terms of the frictional boundary condition at the roll-metal interface. Utilizing simplifying assumptions, an expression results which can then be combined with the equilibrium equations so that analytical expressions for roll force, torque, and extrusion ratio result. Whereas the concept is based on the same physical principles inherent in Orowan’s analysis of rolling, the slab method of analysis used here is less rigorous but has the advantage that analytic expressions are obtainable. Therefore, the equations are applicable for mill control purposes. Furthermore, the analytic expressions can be inverted so that estimates of friction and flow stress may be obtained by roll force and extrusion ratio measurements. In hot rolling, the temperature of the sheet may then be inferred from this value of flow stress, if the material’s flow stress is a known function of temperature and strain rate. Thus, this provides the capability to eventually control sheet temperatures and lubrication conditions by adjusting coolant sprays. Theoretical verification is discussed by comparison with Orowan’s theory and finite element analysis. Accuracy is discussed in terms of hot continuous mill and laboratory reversing mill data.

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