I am interested in shape, dynamics, and mechanics. Often this leads me to problems involving curved thin structures such as chains, strings, rods, membranes, plates, and shells. My group spends much of its time deriving and solving differential equations and playing in the lab.
Much of our recent, current, and planned activity falls into one of the following overlapping areas:
Discontinuities, variational principles, pseudomomentum, and material symmetry.
What balance laws and conserved quantities can help us predict the counterintuitive physics of moving contacts between flexible and rigid bodies? When can we apply action principles to variable-mass problems such as water entry, popping balloons, or the cutting and winding of sheets?
Elasticity, including the behavior of incompatible continua and structures, multi-stability and bifurcation, and the stretching and bending of surfaces.
How do elastic "defects" interact, form patterns, and mediate snap-buckling in plates and shells? When does an elastic strip behave like a rod and when like a plate?
Dynamics of flexible bodies, including their interaction with fluid, and their relative equilibria subject to inertia, drag, and other forces. What shapes are adopted by a towed cable or by unspooling yarn? How do floppy things get caught on other things?
However, our interests are broader than this, and the future is hard to predict.
I am inspired by observation of many engineered structures and aspects of manufacturing related to materials processing, and welcome any excuse to apply geometric tools to nonlinear phenomena in classical continuum mechanics or soft condensed matter.
Here is an incomplete list of recent publications with some comments and informal errata.
I strongly recommend the arXiv versions of the linked articles, as the effects of journal publication often include mutilation of color, layout, and resolution of figures and equations, introduction of copyediting errors into equations and text, and other unfortunate events.
- Exact and approximate mechanisms for pure bending of sheets
Cochleoids and other plane geometry relevant to studies of pleating, folding, creasing, and other forming operations.
A long paper resolving and raising some issues that have been bothering me for a while, and a short paper expanding a point from the long paper, in praise of Biot strains for thin structures (please note that the footnote mentioning neo-Hookean energy is half wrong).
A bendy straw, which can stably adopt just about any one-dimensional shape, only works because it's pre-stressed. Cut one length-wise and watch it relax to a larger radius. An incompatible four-bar linkage works similarly.
A chain dropped onto a table falls faster than it would if the table were not there.
A classical approach to sleeve constraints, peeling, and similar problems.
Fun with SO(3). Multi-stability. Snapping movies. Unexpected problems with strip models.
An old problem revisited. Classification of elastica by momentum and pseudomomentum.
The paper is about contact discontinuities in extended bodies, particularly how the geometry depends on whether you are picking something up or laying it down. Examples include peeling tapes, moored structures, and some musical instruments. The video is an example of two propagating contact discontinuities.
The classical catenaries are a one-parameter family of curves that don't change shape under translation and axial flow. Stick them in a fluid and they do, picking up four additional shape parameters. These solutions include towing, sedimentation, and other problems as subcases.
This note is about a constant arising in dynamical equilibria of thin bodies, which we later learned is related to the Eshelby tensor and to conservation of Kelvin circulation (we now realize that our comparison with the Bernoulli equation is misleading, as it relied on specifics of this problem, and is not generally correct). We also use symmetry arguments to provide the solutions for rotating, flowing strings, such as the steady state of a piece of unspooling yarn.
Phenomena such as kinks, shocks, partial contact, cracking, tearing, and cutting of thin structures involve propagating discontinuities in geometry or applied forces. This was my first clumsy attempt to understand these things in terms of action principles for non-material volumes. Some aspects of the approach probably need correction before it can be extended.
Here we examine the rotation of a flowing, flexible membrane. We find that a minimal model of metrically constrained dynamics generates peaked waves, just like this
In a sheet of paper, a cone is a monopole: a point singularity in intrinsic curvature. So what's a dipole? There are actually two kinds.
At the free end of a moving string, the stress has to vanish, and a result is that the motion of the end is nontrivial. The video is fun and the draft has some new results, but please don’t read it.
These are the three-dimensional "jumpropica" or rigidly rotating strings, generalizing the known planar
Take a pile of chain or rope sitting on a surface, and pull one end really fast. A strange arch-like feature appears. Our paper talks about several interesting issues that came up while we thought about this problem, though these are not necessarily the most important effects behind the arch and related phenomena such as the "chain fountain", about which open questions remain.
Adventures in making 3D curved surfaces by swelling flat 2D sheets.
The original draft had a better title, which you can still see thanks to the arXiv.