Homepage

Shape-programmed shells morph from flat into curved shapes upon stimulation by light, heat, or chemistry. They are ubiquitous throughout biology, and their synthetic counterparts show great promise as soft large-strain actuators. I’ll present several theoretical/computational advances towards assembling a zoo of mechanically strong shape-morphing “mechanisms”. A central theme is encoding Gauss curvature (GC) via patterns of in-plane deformation, due to the mechanical strength inherited by the resultant structures, as Gauss understood centuries ago. The canonical example is a pattern of azimuthal contraction that morphs a planar disk into a cone, which cannot be flattened without energetically costly stretch because its tip bears concentrated GC. In that spirit, I’ll demonstrate novel designs for nematic patterns that encode concentrated GC at generalized “tips” (via topological defects), along ridges (via seams between smooth patterns), and within the central holes of annuli (via spirals). Then, to investigate mechanical strength more quantitatively, we’ll turn to the load-bearing capacity of perfect conical shells. This classical-sounding problem is in fact rather subtle; I will present a new boundary-layer solution, leading to an asymptotic critical force $\propto t^{5/2}$. This surprising scaling is novel, and has broad implications for shell buckling more generally. I also explore deep postbuckling, finding further instabilities producing intricate states with multiple Pogorelov-type curved ridges arranged in concentric circles or Archimedean spirals. Finally, I investigate the forces exerted by such states, which limit lifting performance in shape-morphing cones.

We study the thermal decay of bubble oscillations in an incompressible fluid with surface tension. Particularly, we focus on the isobaric approximation [Prosperetti, JFM, 1991], under which the gas pressure within the bubble is spatially uniform and follows the ideal gas law. This model exhibits a one-parameter family of spherical equilibria, parametrized by the bubble mass. We prove that this family forms an attracting centre manifold for small spherically symmetric perturbations, with solutions converging to the manifold at an exponential rate over time. Furthermore, we show that under either liquid viscosity or irrotational flow assumptions, any equilibrium gas bubble must be spherical by proving that the bubble boundary is a closed surface of constant mean curvature. Additionally, the manifold of spherically symmetric equilibria captures all regular spherically symmetric equilibrium.

We also explore the dynamics of the bubble-fluid system subject to a small-amplitude, time-periodic, spherically symmetric external sound field. For this periodically forced system, we establish the existence of a unique time-periodic solution that is nonlinearly and exponentially asymptotically stable against small spherically symmetric perturbations.

In the latter part of the talk, I will discuss some limitations of the isobaric model in a more general (nonspherically symmetric) irrotational setting. Specifically, I will address issues such as (1) the undamped oscillations of shape modes due to spatial uniformity of the gas pressure, and (2) the incompatibility between viscosity and irroataionality assumptions. Our results suggest that to accurately capture the effect of thermal damping on the dynamics of general deformations of a gas bubble, the model should be considered within a framework that includes either non-zero vorticity, corrections to the isobaric approximation, or both.

If time permits, I will present ongoing work on the existence of nonspherically symmetric equilibrium bubbles in a rotational framework.

This talk is based on joint work with Michael I. Weinstein ([Arch. Ration. Mech. Anal. 2023], [Nonlinear Anal. 2024], [arXiv:2408.03787], and work in progress).

TBA

TBA