Self-Ordering of Buckling, Bending, and Bumping Beams

Physical Review Letters · 2023-04-03 · 18 citations

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 confined elastic beams. Using tabletop experiments, simulations, and standard theory from statistical mechanics, we predict the amount of confinement (growth or 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 metamaterial are 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 metamaterial, with a tunable resistance to compressive force.

Wrinkling and developable cones in centrally confined sheets

Physical review. E · 2023-09-11 · 4 citations

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 simulations, we probe the confinement response of circular sheets, flattened in their central region and quasistatically 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 paper 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.

Wrinkling and developable cones in centrally confined sheets

PubMed · 2022-09-13

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 simulations, we probe the confinement response of circular sheets, flattened in their central region and quasistatically 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 paper 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.

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

medRxiv · 2021-04-07 · 17 citations

Abstract 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 and Bush, 2021). Here, we rephrase this safety guideline in terms of occupancy time and mean exhaled carbon dioxide concentration in an indoor space, thereby enabling the use of CO 2 monitors in the risk assessment of airborne transmission of respiratory diseases. While CO 2 concentration is related to airborne pathogen concentration (Rudnick and Milton, 2003), 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 persons 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 CO 2 measurements. Illustrative examples of implementing our guideline are presented using data from CO 2 monitoring in university classrooms and office spaces. Impact Statement There is mounting scientific evidence that COVID-19 is primarily transmitted through indoor airborne transmission, as arises when a susceptible person inhales virus-laden aerosol droplets exhaled by an infectious person. A safety guideline to limit indoor airborne transmission (Bazant and Bush, 2021) has recently been derived that complements the public health guidelines on surface cleaning and social distancing. We here recast this safety guideline in terms of total inhaled carbon dioxide, as can be readily monitored in most indoor spaces. Our approach paves the way for optimizing air handling systems by balancing health and financial concerns, informs policy for safely re-opening schools and businesses as the pandemic runs its course, and may be applied quite generally in the mitigation of airborne respiratory illnesses, including influenza.

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

Flow · 2021 · 62 citations

• Environmental science
• Environmental health
• Engineering

Abstract 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 ( ${\rm CO}_{2}$ ) concentration in an indoor space, thereby enabling the use of ${\rm CO}_{2}$ monitors in the risk assessment of airborne transmission of respiratory diseases. While ${\rm CO}_{2}$ 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 ${\rm CO}_{2}$ measurements. Illustrative examples of implementing our guideline are presented using data from ${\rm CO}_{2}$ monitoring in university classrooms and office spaces.

Dynamic Buckling of an Elastic Ring in a Soap Film

Physical Review Letters · 2020 · 36 citations

• Mechanics
• Materials science
• Physics

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 on floating thin elastic sheets: A mathematical model

Physical Review Fluids · 2020-01-27 · 9 citations

Ballistic impact can be used to induce dynamic wrinkling in a floating elastic sheet. A mathematical model is developed and analyzed to describe the resulting dynamics: an elastocapillary wave propagates out from the point of impact, and the evolving wrinkle wavelength is dictated by fluid inertia.

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

Physical review. E · 2020-05-13 · 26 citations

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 postbifurcation 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 postbifurcation 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.

On the figure of elastic planets I: gravitational collapse and infinitely many equilibria

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2019-04-01 · 4 citations

A classic problem of elasticity is to determine the possible equilibria of an elastic planet modelled as a homogeneous compressible spherical elastic body subject to its own gravitational field. In the absence of gravity, the initial radius is given and the density is constant. With gravity and for small planets, the elastic deformations are small enough so that the spherical equilibria can be readily obtained by using the theory of linear elasticity. For larger or denser planets, large deformations occur and the general theory of nonlinear elasticity is required to obtain the solution. Depending on the elastic model, we show that there may be parameter regimes where there exist no equilibrium or arbitrarily many equilibria. Yet, at most two of them are dynamically stable with respect to radial disturbances. In some of these models, there is a critical initial radius at which spherical solutions cease to exist. For planets with larger initial radii, there is no spherical solution as the elastic forces are not sufficient to balance the gravitational force. Therefore, the system undergoes gravitational collapse, an unexpected phenomenon within the framework of classical continuum mechanics.

Dynamic buckling of rings and annuli

Bulletin of the American Physical Society · 2019-03-08