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Metal rolling is a widespread and well-studied process, and many finite-element (FE) rolling simulations can be found in the scientific literature. However, these FE simulations are typically limited in their resolution of through-thickness variations. In this paper, we carefully assess the accuracy of a number of FE approaches, and find that at least 60 elements through-thickness are needed to properly resolve through-thickness variation; this is significantly more than is used elsewhere in the metal rolling literature. In doing so, we reveal an oscillatory stress pattern, which is not usually observed in simulations but which we can validate by comparison with recent analytical work, and which is completely deterministic, not arising from numerical noise or error. We show that these oscillations contribute to the formation of residual stress and may help predict curvature in asymmetric rolled sheets, a phenomenon which is currently not well understood. Accurate through-thickness variation of stress and strain would also have implications for modelling microstructure evolution, damage, and surface finish., Comment: 33 pages, 16 figures

A new semi-analytic model of the metal rolling process is presented and validated against finite element simulations. The model generalises the classical slab method of modelling cold rolling, and 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 the asymptotic method of multiple scales, with the systematic assumptions of a long thin roll gap and a comparably small Coulomb friction coefficient. The leading-order solution varies only on a long length scale corresponding to the roll-gap length and matches with slab methods. The next-order correction varies on both this long length scale and a short length scale 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. Matlab code for evaluating the model is provided in the supplementary material., Comment: 32 pages, 10 figures

Mean-field equations have been developed recently to approximate the dynamics of the Deffuant model of opinion formation. These equations can describe both fully-mixed populations and the case where individuals interact only along edges of a network. In each case, interactions only occur between individuals whose opinions differ by less than a given parameter, called the confidence bound. The size of the confidence bound parameter is known to strongly affect both the dynamics and the number and location of opinion clusters. In this work we carry out a mathematical analysis of the mean-field equations to investigate the role of the confidence bound and boundaries on these important observables of the model. We consider the limit in which the confidence bound interval is small, and identify the key mechanisms driving opinion evolution. We show that linear stability analysis can predict the number and location of opinion clusters. Comparison with numerical simulations of the model illustrates that the early-time dynamics and the final cluster locations can be accurately approximated for networks composed of two degree classes, as well as for the case of a fully-mixed population.

Dynamic buckling may occur when a load is rapidly applied to, or removed from, an elastic object at rest. In contrast to its static counterpart, dynamic buckling offers a wide range of accessible patterns depending on the parameters of the system and the dynamics of the load. To study these effects, we consider experimentally the dynamics of an elastic ring in a soap film when part of the film is suddenly removed. The resulting change in tension applied to the ring creates a range of interesting patterns that cannot be easily accessed in static experiments. Depending on the aspect ratio of the ring's cross section, high-mode buckling patterns are found in the plane of the remaining soap film or out of the plane. Paradoxically, while inertia is required to observe these non-trivial modes, the selected pattern does not depend on inertia itself. The evolution of this pattern beyond the initial instability is studied experimentally and explained through theoretical arguments linking dynamics to pattern selection and mode growth. We also explore the influence of dynamic loading and show numerically that by imposing a rate of loading that competes with the growth rate of instability, the observed pattern can be selected and controlled., Comment: 6 pages, 5 figures

We investigate impact of a sphere onto a floating elastic sheet and the resulting formation and evolution of wrinkles in the sheet. Following impact, we observe a radially propagating wave, beyond which the sheet remains approximately planar but is decorated by a series of radial wrinkles whose wavelength grows in time. We develop a mathematical model to describe these phenomena by exploiting the asymptotic limit in which the bending stiffness is small compared to stresses in the sheet. The results of this analysis show that, at a time $t$ after impact, the transverse wave is located at a radial distance $r\sim t^{1/2}$ from the impactor, in contrast to the classic $r\sim t^{2/3}$ scaling observed for capillary--inertia ripples produced by dropping a stone into a pond. We describe the shape of this wave, starting from the simplest case of a point impactor, but subsequently addressing a finite-radius spherical impactor, contrasting this case with the classic Wagner theory of impact. We show also that the coarsening of wrinkles in the flat portion of the sheet is controlled by the inertia of the underlying liquid: short-wavelength, small-amplitude wrinkles form at early times since they accommodate the geometrically-imposed compression without significantly displacing the underlying liquid. As time progresses, the liquid accelerates and the wrinkles grow larger and coarsen. We explain this coarsening quantitatively using numerical simulations and scaling arguments, and we compare our predictions with experimental data., Comment: 30 pages, 9 figures. Small edits toaccepted version

The wrinkling of thin elastic objects provides a means of generating regular patterning at small scales in applications ranging from photovoltaics to microfluidic devices. Static wrinkle patterns are known to be governed by an energetic balance between the object's bending stiffness and an effective substrate stiffness, which may originate from a true substrate stiffness or from tension and curvature along the wrinkles. Here we investigate dynamic wrinkling, induced by the impact of a solid sphere onto an ultra-thin polymer sheet floating on water. The vertical deflection of the sheet's centre induced by impact draws material radially inwards, resulting in an azimuthal compression that is relieved by the wrinkling of the entire sheet. We show that this wrinkling is truly dynamic, exhibiting features that are qualitatively different to those seen in quasi-static wrinkling experiments. Moreover, we show that the wrinkles coarsen dynamically because of the inhibiting effect of the fluid inertia. This dynamic coarsening can be understood heuristically as the result of a dynamic stiffness, which dominates the static stiffnesses reported thus far, and allows new controls of wrinkle wavelength., Comment: 8 pages, 4 figures. Please see published version for supplementary movies and SI Appendix