Through-Thickness Modelling of Metal Rolling using Multiple-Scale Asymptotics

TL;DR

This paper introduces a semi-analytic multiple-scales asymptotic model for metal rolling that predicts through-thickness stress and strain oscillations efficiently, matching finite element simulations despite simplified assumptions.

Contribution

The paper presents the first semi-analytic model capable of predicting through-thickness oscillations in metal rolling using multiple-scales asymptotics, with validation against finite element results.

Findings

Model predicts rapid stress and strain oscillations in the workpiece.

Matches well with finite element simulations despite simplified assumptions.

Computationally efficient, taking only seconds to evaluate.

Abstract

A new semi-analytic model of the metal rolling process is introduced, which, for the first time, is able to predict the through-thickness stress and strain oscillations present in long thin roll-gaps. The model is based on multiple-scales asymptotics, assuming a long thin roll-gap and a comparably small Coulomb friction coefficient. The leading-order solution varies only on a long lengthscale corresponding to the roll-gap length and matches with slab models. The next-order correction varies on both this long lengthscale and a short lengthscale associated with the workpiece thickness, and reveals rapid stress and strain oscillation both in the rolling direction and through the thickness. For this initial derivation, the model assumes a rigid perfectly-plastic material behaviour. Despite these strong assumptions, this model compares well with finite element simulations that employ more realistic material behaviour (including elasticity and strain hardening). These assumptions facilitate the simplest possible model to provide a foundational understanding of the complex through-thickness behaviour observed in the finite element simulations, while requiring an order of only seconds to compute. This model can form the foundation of further improved models with more complicated mechanics in the future. Matlab code for evaluating the model is provided in the supplementary material.