Dynamic buckling of an inextensible elastic ring: Linear and nonlinear analyses

Slender elastic objects such as a column tend to buckle under loads. While static buckling is well understood as a bifurcation problem, the evolution of shapes during dynamic buckling is much harder to study. Elastic rings under normal pressure have emerged as a theoretical and experimental paradigm for the study of dynamic buckling with controlled loads. Experimentally, an elastic ring is placed within a soap film. When the film outside the ring is removed, surface tension pulls the ring inward, mimicking an external pressurization. Here we present a theoretical analysis of this process by performing a post-bifurcation analysis of an elastic ring under pressure. This analysis allows us to understand how inertia, material properties, and loading affect the observed shape. In particular, we combine direct numerical solutions with a post-bifurcation asymptotic analysis to show that inertia drives the system towards higher modes that cannot be selected in static buckling. Our theoretical results explain experimental observations that cannot be captured by a standard linear stability analysis.Comment: 18 pages, 10 figure

Lubricated wrinkles: imposed constraints affect the dynamics of wrinkle coarsening

We study the dynamic coarsening of wrinkles in an elastic sheet that is compressed while lying on a thin layer of viscous liquid. When the ends of the sheet are instantaneously brought together by a small distance, viscous resistance initially prevents the sheet from adopting a globally buckled shape. Instead, the sheet accommodates the compression by wrinkling. Previous scaling arguments suggested that a balance between the sheet's bending stiffness and viscous effects lead to a wrinkle wavelength $\lambda$ that increases with time $t$ according to $\lambda\propto t^{1/6}$. We show that taking proper account of the compression constraint leads to a logarithmic correction of this result, $\lambda\propto (t/\log t)^{1/6}$. This correction is significant over experimentally observable time spans, and leads us to reassess previously published experimental data.Comment: 12 pages. Version accepted in Phys. Rev. Fluids (with small correction to bibliography

Self-Ordering of Buckling, Bending, Bumping Beams

A collection of thin structures buckle, bend, and bump into each-other when confined. This contact can lead to the formation of patterns: hair will self-organize in curls; DNA strands will layer into cell nuclei; paper, when crumpled, will fold in on itself, forming a maze of interleaved sheets. This pattern formation changes how densely the structures can pack, as well as the mechanical properties of the system. How and when these patterns form, as well as the force required to pack these structures is not currently understood. Here we study the emergence of order in a canonical example of packing in slender-structures, i.e. a system of parallel growing elastic beams. Using experiments, simulations, and simple theory from statistical mechanics, we predict the amount of growth (or, equivalently, the amount of compression) of the beams that will guarantee a global system order, which depends only on the initial geometry of the system. Furthermore, we find that the compressive stiffness and stored bending energy of this meta-material is directly proportional to the number of beams that are geometrically frustrated at any given point. We expect these results to elucidate the mechanisms leading to pattern formation in these kinds of systems, and to provide a new mechanical meta-material, with a tunable resistance to compressive force

Dynamics of wrinkling in ultrathin elastic sheets

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 Appendi

Impact on floating thin elastic sheets: A mathematical model

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 versio

Wrinkling and developable cones in centrally confined sheets

Thin sheets respond to confinement by smoothly wrinkling, or by focusing stress into small, sharp regions. From engineering to biology, geology, textiles, and art, thin sheets are packed and confined in a wide variety of ways, and yet fundamental questions remain about how stresses focus and patterns form in these structures. Using experiments and molecular dynamics (MD) simulations, we probe the confinement response of circular sheets, flattened in their central region and quasi-statically drawn through a ring. Wrinkles develop in the outer, free region, then are replaced by a truncated cone, which forms in an abrupt transition to stress focusing. We explore how the force associated with this event, and the number of wrinkles, depend on geometry. Additional cones sequentially pattern the sheet, until axisymmetry is recovered in most geometries. The cone size is sensitive to in-plane geometry. We uncover a coarse-grained description of this geometric dependence, which diverges depending on the proximity to the asymptotic d-cone limit, where the clamp size approaches zero. This work contributes to the characterization of general confinement of thin sheets, while broadening the understanding of the d-cone, a fundamental element of stress focusing, as it appears in realistic settings.Comment: 11 pages, 9 figure

Dynamic buckling instabilities in fluids and solids

Many natural phenomena encountered in nature may be understood through the paradigm of buckling instability. Examples include the design of columns in structural engineering, the folding of geological formations, the collapse of blood vessels, and the fragmentation of uncooked spaghetti, to name only a few. The phenomenon of buckling has traditionally been seen only as a nuisance, but more recently it has also proven useful as a potential tool for pattern formation, particularly at small scales. Many studies have focused on the features of static buckling, particularly pattern formation. This thesis focuses on how dynamic buckling affects the spontaneous selection of patterns in a number of canonical problems. We first study the effect of curvature on dynamic buckling in circular geometries. We study the pattern observed when a ring made of either a viscous liquid or an elastic solid is subject to a suddenly applied external pressure. We develop numerical schemes to study the evolution of the ring’s profile and compare those results to linear stability analysis. A weakly nonlinear analysis is performed to understand the behaviour of the observed shape beyond the onset of instability, with results that compare well with both numerical and experimental work. In the late stages of instabilities, buckles merge — the wrinkle pattern coarsens. To understand this coarsening we consider the non inertial problem of a beam sit- ting on a viscous layer. We explain how wrinkle coarsening occurs as the result of wave dispersion and quantify coarsening by presenting an asymptotic analysis of the governing equation. We also study the effect of confinement and show that it has a crucial effect on coarsening at late times. We develop an analytical expression for the effect of confinement, which allows us to better explain previously published experimental data. Finally, our numerical solution reveals a new regime as the wavelength of wrinkles become comparable to the size of the system. In this regime the dynamics is relatively slow with the system remaining for long periods with one particular mode before rapidly switching to another. We study this regime, the so-called temporal cascade, in detail. The problems we are considering have different geometries and different material constitutive relations, but highlight a number of common themes. Mathematically, those systems are described by a Partial Differential Equation with a global constraint whose numerical solution is found by framing it in the language of Differential Algebraic Equations

Monitoring carbon dioxide to quantify the risk of indoor airborne transmission of COVID-19

A new guideline for mitigating indoor airborne transmission of COVID-19 prescribes a limit on the time spent in a shared space with an infected individual (Bazant & Bush, Proceedings of the National Academy of Sciences of the United States of America, vol. 118, issue 17, 2021, e2018995118). Here, we rephrase this safety guideline in terms of occupancy time and mean exhaled carbon dioxide (CO2) concentration in an indoor space, thereby enabling the use of CO2 monitors in the risk assessment of airborne transmission of respiratory diseases. While CO2 concentration is related to airborne pathogen concentration (Rudnick & Milton, Indoor Air, vol. 13, issue 3, 2003, pp. 237–245), the guideline developed here accounts for the different physical processes affecting their evolution, such as enhanced pathogen production from vocal activity and pathogen removal via face-mask use, filtration, sedimentation and deactivation. Critically, transmission risk depends on the total infectious dose, so necessarily depends on both the pathogen concentration and exposure time. The transmission risk is also modulated by the fractions of susceptible, infected and immune people within a population, which evolve as the pandemic runs its course. A mathematical model is developed that enables a prediction of airborne transmission risk from real-time CO2 measurements. Illustrative examples of implementing our guideline are presented using data from CO2 monitoring in university classrooms and office spaces

Dynamic buckling of an elastic ring in a soap film

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 nontrivial 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

Impact of Three-Year Intermittent Preventive Treatment Using Artemisinin-Based Combination Therapies on Malaria Morbidity in Malian Schoolchildren

Previous studies have shown that a single season of intermittent preventive treatment in schoolchildren (IPTsc) targeting the transmission season has reduced the rates of clinical malaria, all-cause clinic visits, asymptomatic parasitemia, and anemia. Efficacy over the course of multiple years of IPTsc has been scantly investigated. Methods: An open, randomized-controlled trial among schoolchildren aged 6–13 years was conducted from September 2007 to January 2010 in Kolle, Mali. Students were included in three arms: sulphadoxine-pyrimethamine+artesunate (SP+AS), amodiaquine+artesunate (AQ+AS), and control (C). All students received two full doses, given 2 months apart, and were compared with respect to the incidence of clinical malaria, all-cause clinic visits, asymptomatic parasitemia, and anemia. Results: A total of 296 students were randomized. All-cause clinic visits were in the SP+AS versus control (29 (20.1%) vs. 68 (47.2%); 20 (21.7%) vs. 41 (44.6%); and 14 (21.2%) vs. 30 (44.6%); p < 0.02) in 2007, 2008, and 2009, respectively. The prevalence of asymptomatic parasitemia was lower in the SP+AS compared to control (38 (7.5%) vs. 143 (28.7%); and 47 (12.7%) vs. 75 (21.2%); p < 0.002) in 2007 and 2008, respectively. Hemoglobin concentration was significantly higher in children receiving SP+AS (11.96, 12.06, and 12.62 g/dL) than in control children (11.60, 11.64, and 12.15 g/dL; p < 0.001) in 2007, 2008, and 2009, respectively. No impact on clinical malaria was observed. Conclusion: IPTsc with SP+AS reduced the rates of all-cause clinic visits and anemia during a three-year implementation