Mathematical modelling of the cold-rolling process

In this thesis two types of mathematical modelling techniques are used to model metal sheet rolling: the finite element (FE) method and asymptotics-based approaches. The modelling conducted here forms part of a large-scale research project aimed at improving the efficiency and control capabilities of the rolling process. Since large amounts of CO2 are produced globally each year during rolling, it is of utmost importance to continue to improve the fundamental understanding of such metal forming processes in order to reduce waste. In the absence of accurate experimental descriptions of through-thickness stress and strain distributions, the FE model in this thesis provides a necessary benchmark for asymptotics-based models. Given their quick-to-compute nature, asymptotics-based models provide a potential route to online control of the rolling process, if sufficient accuracy with respect to the FE method can be achieved.

In the FE analysis of the rolling process in this thesis, the model is carefully developed to give accurate through-thickness predictions of stress and strain distributions during the steady-state cold-rolling process. These through-thickness predictions unveil an oscillatory pattern, which is largely absent in the existing literature, that is shown to have important consequences for residual stress in the rolled sheet. Care is taken by considering the number of elements through thickness, convergence to a steady state, and the avoidance of other numerical artefacts such as shear locking and hourglassing. We find that the meshes used in previous FE models of cold rolling are usually woefully under-resolved, which reduces FE model accuracy in a number of ways. Convergence of roll force and roll torque, used in previous studies to validate models, are shown here to be poor indicators of through-thickness stress and strain convergence. We also show that the through-thickness oscillatory pattern may have important consequences for residual stress predictions and for predicting the curvature of the sheet during asymmetric rolling.

In terms of the asymptotic analysis, we begin by considering simple one-dimensional models of the full rolling problem, which includes a rigid-perfectly-plastic analysis inside the roll gap and a linear elastic analysis outside the roll gap. This asymptotics-based model compares reasonably well against FE results for long-and-thin roll gaps since through-thickness variations are less important for these parameter regimes. However, the one-dimensional model performs poorly otherwise, as it fails to capture the oscillatory pattern observed in the FE results. We conduct two-dimensional boundary-layer analysis around the roll-gap entrance to describe the rapid change in stress and strain quantities in this region of the deforming sheet. We also predict the boundary between the sub-yield and at-yield zones in the roll-gap entrance.

Although FE models are quite accurate, their slow computation time does not allow for real-time control. Accurate asymptotics-based models such as the rigid-perfectly-plastic analysis conducted by Erfanian et al. (2024) (which captures the oscillatory pattern observed in the FE results) are desirable due to their quick-to-compute nature. These asymptotic models could prove to be crucial for online control to correct for manufacturing errors as they happen.