Schematics of rolling of metal and pasta sheets. There is a thin horizontal sheet and long thin rollers are placed above and below the sheet. As the rollers turn, the sheet moves between them and is squeezed to reduce its thickness.

Limerick. Mathematical modelling reveals stress distributions in metal sheets

Featured, industry 4.0, Limerick

By Doireann O’Kiely, Frank Flanagan, Mozadheh Erfanian, Edward James Brambley, Omer Music & Alison O’Connor

Metal sheet rolling is a manufacturing process used to change the shape and grain structure of sheets of steel and aluminium.  The set-up is remarkably like the pasta machines used to make sheets of lasagne: a sheet is fed between two rollers whose spacing is slightly less than the thickness of the sheet.  As the rollers turn, they squeeze the sheet, reducing its thickness (Figure 1).  Manufacture of primary materials such as steel and aluminium is a major contributor to global CO2 emissions, so processing metal sheets efficiently, effectively, and with minimal waste, is environmentally important.

Schematics of rolling of metal and pasta sheets. There is a thin horizontal sheet and long thin rollers are placed above and below the sheet. As the rollers turn, the sheet moves between them and is squeezed to reduce its thickness.

Figure 1: Schematic of rolling process [1]; pasta machine for rolling lasagne sheets [2]

Assuming deformation and stresses are uniform across the sheet yields slab models, which are mathematically tractable but lack detail.  Finite element modelling gives a more detailed picture but careful implementation is crucial.  A recent collaboration between MACSI researchers at the University of Limerick and colleagues at the University of Warwick and University of Cambridge has used asymptotic and finite element approaches in harmony to improve both analytical and numerical descriptions of the process.  This has yielded intriguing new insights into the stresses and dynamics inside a metal sheet during the rolling process. 

At relatively small stresses, metal undergoes elastic deformation.  This is reversible, so objects return to their original shape when the stress is removed.  However, during sheet rolling, stresses exceed the metals yield point, causing permanent deformation.  This deformation inside this roll gap can be approximately described with the equations of plastic flow.  When the metal sheet touches the rollers, friction causes a shear stress which pulls the sheet between the rollers.  The volume of the sheet does not change (apart from a small elastic strain), so as the sheet thickness decreases inside the roll gap, its speed must increase.  If its speed exceeds the roller speed at some point, it then experiences a shear stress in the opposite direction, opposing its motion as it exits the roll gap (illustrated by a colour change in Figure 2).

• Figure 2: Colour plots of shear stress as a function of position in the sheet, predicted by numerical and asymptotic methods. The pattern has lobes of high stress that repeat periodically through the sheet. In the numerical solution the shear lobes are smooth in shape, while in the asymptotic prediction they have sharp corners.

Figure 2:  Shear stress in a metal sheet during rolling described by finite elements (left, [3]) and multiple-scales analysis (right, [4]), for a roll gap aspect ratio 0.075.

Finite element simulations were performed in Abaqus, with particular care taken to ensure that spatial variations were fully resolved.  This required a significantly finer mesh than is usually used, but revealed an intriguing pattern of shear lobes inside the sheet, where the metal experiences oscillatory stresses (Figure 2).  This is reflected in the velocity of the sheet, which cycles between velocity plateaus and regions of rapid acceleration, rather than the smooth acceleration that might have been expected (Figure 3).

• Figure 3: Plot of the metal velocity at the centre and the surface of a metal sheet during rolling. The surface velocity increases when the sheet first touches the rollers, then platueaus, and then increases again. At the centre the velocity does not increase until the stress wave in Figure 2 reaches the sheet centre. This pattern is repeated along the metal sheet.

Figure 3: Metal velocity as a function of distance travelled through the roll gap.  The velocity increases rapidly, plateaus, and then increases rapidly again, in line with the shear oscillations in Figure 2.

The aspect ratio of the thickness to length of the roll gap is typically small, and this small parameter can be used to facilitate further mathematical modelling.  However, the pattern of shear lobes points to two distinct horizontal lengthscales: one associated with the large roll gap length, and a second associated with the small roll gap thickness.  For this reason, a multiple scales analysis is required.  This analysis reveals that the stress and displacements take the form of waves, which initiate at the sheet surface when it first touches the rollers, and then bounce over and back across the sheet.  The wave amplitudes are governed by a Burgers equation, with expansion fans and shocks occurring in the mathematical model whose smoother remnants are also visible in numerical solutions (Figure 3).

Currently, investigations are ongoing into an interesting previously-unseen correlation between the stress pattern within the roll gap and residual stresses in the rolled sheet.  Control over these highly-localized residual stresses could have industrial applications if better understood and quantified, because residual stresses can cause unwanted distortion and springback immediately after rolling and in subsequent production processes.