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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).

Liquid crystal elastomers (LCEs) marry the large deformation response of a cross-linked polymer network with the nematic order of liquid crystals pendent to the network. When a monodomain LCE sheet (of uniform nematic order) is clamped and stretched, it displays a soft elastic response dictated by fine-scale oscillations in the nematic order. Alternatively, when a pattern LCE sheet is heated, it actuates into a complex shape dictated by how the nematic order is programmed. This talk explains both phenomenon in the context of asymptotic analysis of a widely used bulk (3D) elastic theory for these material as the sheet thickness tends to zero. In the first part of the talk, we derive the membrane theory for LCEs as the Γ-convergence of the bulk energy normalized by the thickness, and we show how it explains the soft elasticity and fine-scale microstructure in stretched sheets. In the second part of the talk, we derive the bending theory for LCE sheets as a Γ-convergence of the bulk energy normalized by the thickness cubed, and we show how it explains and characterizes programmable actuation.

Compliant shell mechanisms are creased and corrugated thin-walled structures that can drastically change shape to move and deploy. Ideally, thin shells deform isometrically. The problem of finding, or disproving the existence of, isometric deformations for various surfaces preoccupied many mathematicians and mechanicians. The most noteworthy results undoubtedly pertain to three broad categories of surfaces: developable, convex, and axisymmetric. In the modern context of computer graphics, discrete differential geometry and “Origami science,” more focus has been directed towards tri- and quad-based polyhedral surfaces. In this talk, we report on recent progress regarding yet another class of surfaces: periodic surfaces, i.e., surfaces that are invariant by a two-dimensional lattice of translations. In the spirit of the theory of homogenization, focus is on asymptotic isometries that survive a passage to the limit in the lattice constant. Various examples and illustrations are presented.

The uses of graphene in microscopic and mesoscopic elastic meta-materials has generated a great deal of interest in the effects of temperature on the elastic properties of atomically-thin/quasi-2-D materials. Whereas for 1-D elastic polymers, no ordered phase exists and thus the polymer performs a self-avoiding random walk, 2-D membranes have an ordered flat phase. The talk will focus on the application of the renormalization group within the ordered flat phase. When the reference state is flat, temperature-generated wrinkles induce an “effective” change/renormalization such that the bending rigidity diverges with increasing system size and the Young’s modulus converges to zero with power law exponents. In addition, these exponents are not independent of one another due to a Ward identity, from the language of field theory (in other words, as derived from a symmetry of the Hamiltonian). Ample discussion will be about the method and idea of the renormalization group itself.

Networks of coupled oscillators have been used to model various complex phenomena ranging from circadian rhythms to high-voltage power grids. Recently, many researchers have sought to understand the behaviour of such systems when the size of the network becomes very large, with special attention given to randomly generated networks. These systems are governed by a large number of coupled ODEs and it is often convenient to study them by first understanding their behaviour in an appropriate continuum limit. To this end, graphon theory has proven to be a helpful tool. A graphon is a symmetric kernel on the unit square that has two interpretations: First, they can be understood as the large size limit of sequences of graphs in an appropriate topology. Second, they serve as a model for generating finite random graphs. We use these ideas to show that for a sequence of size-n random graphs, W_n generated from a given graphon, W, solutions to the system of n ODEs governed by W_n converge to solutions of an integro-differential equation governed by W in the continuum limit. We use this convergence result to prove synchronization for a well-known class of coupled oscillator models and relate our results to the preexisting body of literature on global stability of synchronized states on random networks. This is joint work with Shriya Nagpal, Francesca Parise and Steven Strogatz.