Investigation Into Multi-Scale Contact Mechanics Behaviors in the Cold Rolling of Metal Strip and Sheet Using a Novel Stochastic Roll-Stack Modeling Approach

In the cold rolling of thin metal strip and sheet, both the macro-scale and micro-scale dimensional quality are known to be strongly influenced by deviations in the diameter and surface condition of the work rolls. Furthermore, high-resolution industrial measurements of work rolls reveal that the magnitude of diameter deviations vary significantly along the roll axis length, as well as across samples, and during rolling. This circumstance implies that analysis of the effects of work-roll diameter deviations on the strip or sheet dimension quality requires a multi-dimensional random field modeling approach. Conventional roll-stack deflection models, however, are either too computationally expensive or exploit too many simplifying assumptions to allow them to be used for rigorous stochastic analysis, even at the macro-scale. Moreover, conventional modeling to examine the influence of micro-scale (roughness) surface conditions of the work rolls has only been carried out for two dimensional studies, hence neglecting any possible coupling effects between the micro-scale surface asperity contact mechanics and the macro-scale or three-dimensional bulk-body roll-stack deformations. Described in this work is a stochastic, simplified-mixed finite element method (SSM-FEM) that can efficiently predict the coupling effects between non-symmetric, macroscale roll-stack deformation mechanics and micro-scale surface roughness. The method thus captures multi-scale geometry deviations of work rolls, including diameter deviations along the roll axis as well as varying high-fidelity surface roughness distributions. The SSM-FEM approach developed also accommodates three-dimensional bending, shearing, and flattening type bulk-body roll-stack deformations under binary nonlinear contact conditions. The nonlinearities arise from elastic-plastic material behavior of both the rolls and the rolled strip or sheet, as well as the changing micro-scale and macro-scale contact conditions along the roll length. Performance of the multi-scale SSM-FEM approach is assessed first at the macro-scale for continuously variable crown (CVC) mills using a basic theoretical validation, several comparisons to large-scale finite element solutions, and validation against industrial measurements from a production 6-high mill. In addition, three-dimensional studies using the SSM-FEM approach on both 4-high and 6-high rolling mills reveal new multi-scale coupling effects, including non-uniform roughness transfer characteristics from the work-rolls to the sheet, as well as perturbations to the sheet thickness profile (or ”crown”) and the corresponding contact force distribution between the work-roll and the sheet. Finally, a stochastic analysis of the work roll diameter deviations when characterized as a random field rather than as independent random variations as is conventionally done, show that the reliability of the shape/flatness of the rolled sheet is strongly affected by the non-proximal correlations between the diameter deviations. Studies are provided for work rolls having residual roll grinding errors in different operation conditions, including new, warm, and worn states. These studies, incorporating the new SSM-FEM approach, provide new understanding into the to relationship between the roll grinding precision and the subsequent dimensional quality of the rolled metal sheet. The described multi-scale SSM-FEM approach is general, and applies to any multi-scale contact problem involving plates and/or shear-deformable beams having multiple contact interfaces and arbitrary surface profile geometries.