Strip flatness prediction in cold rolling: a review of models, challenges, and emerging directions

Cold rolling is one of the most widely applied metal-forming processes, where strip flatness is a critical quality requirement affecting downstream manufacturability, dimensional accuracy and product performance. Flatness defects arise from non-uniform longitudinal stresses generated through tightly coupled interactions between roll bite mechanics, roll stack deformation, friction and lubrication, material behaviour and post-bite buckling. Although numerous modelling strategies exist, the literature remains fragmented across analytical, numerical, hybrid and data driven approaches, making it difficult to identify the most appropriate tools for prediction and control. This review provides a unified framework for strip flatness prediction in cold rolling, presenting a structured taxonomy of modelling approaches and evaluating their assumptions, predictive capability, computational cost and industrial applicability. Analytical and semi-analytical models are compared with numerical techniques and hybrid coupled formulations. Key transversal components, including constitutive laws, friction and lubrication models and buckling formulations, are examined with respect to their influence on widthwise stress distributions and the formation of flatness defects. Recent developments in physics-based, data driven and hybrid modelling, including digital twins and reduced-order models, are discussed with emphasis on their potential to meet the real-time requirements of advanced flatness-control systems. Major research gaps are identified in mixed-lubrication modelling, advanced constitutive descriptions, transient buckling prediction, standardised validation benchmarks and hybrid physics–data driven modelling to support digital twin deployment. By synthesising current knowledge and highlighting emerging research directions, this review provides guidance for selecting and developing strip modelling strategies capable of supporting high precision crown and flatness control in modern, data-enriched rolling mills.