Understanding the mechanics of thin elastic rods under geometrically nonlinear configurations is a major thrust of our research. There is a recognized importance of filamentlike structures across a wide range of length scales: from the coiling of nanotubes to transoceanic cables and pipelines. The postbuckling regime of slender elastic rods is typically underpinned by a complex energy landscape, hystheresis, and multistability. As such, closed analytical solutions are often out of reach and wellestablished computational techniques (e.g., the finite element method) can be limited in these scenarios. We seek to circumnavigate these challenges by gaining physical insight from precision model experiments with elastic rods, coupled with scaling analyses and numerical tools, the discrete elastic rod (DER) method, that we have ported from the Computer Graphics community in collaboration with Eitan Grinspun (Columbia University).
Topics that we have investigated on the mechanics of thin rods include: mechanics of elastic knots, coiling patterns of a thin rod deployed onto a rigid substrate, buckling of coiled tubing injected into horizontal wellbores, deformation and buckling of helical rods rotating in a viscous fluid and the mechanics of curly hair. A more detailed account of these examples and other problems is provided below.
Patterns of carbon nanotubes by flowdirected deposition on substrates with architectured topographies with: M. K. Jawed, N. G. Hadjiconstantinou and D. M. Parks 


Publications: 
Instabilities of a flexible helical rod rotating in a viscous fluid with: Khalid Jawed 


We also consider the case when the helical filament is simultaneously subjected to an axial flow. Under axial flow, and in the absence of rotation, the initially helical rod is extended. Above a critical flow speed its configuration comprises a straight portion connected to a localized helix near the free end. When the rod is also rotated about its helical axis, propulsion is only possible in a finite range of angular velocity, with an upper bound that is limited by buckling of the soft helix arising due to viscous stresses. A systematic exploration of the parameter space allows us to quantify regimes for successful propulsion for a number of specific bacteria. Publications: 
The interplay between the mechanics and topology of elastic knots with: Khalid Jawed, Peter Dieleman and Basile Audoly 


Publications: Press Coverage: 
Buckling of a rod inside a cylindrical constraint: Applications to coiled tubing operations with: Jay Miller, Jahir Prabon and Nathan Wicks 


Image courtesy of SchlumbergerDoll Research. Publications: Press Coverage: 
Coiling ‘spaghetti’ onto rigid substrates with: Eitan Grinspun 


In this study, we combine precision model experiments with computer simulations tools and explore the mechanics of coiling. In particular, the natural curvature of the rod is found to dramatically affect the coiling process. We have introduced a computational framework that is widely used in computer animation into engineering, as a predictive tool for the mechanics of filamentary structures. This work was done in close collaboration with Eitan Grinspun’s Computer Graphics Group (Columbia University). [Introductory video about this study] [Video showing a detailed comparison between Experiments and Simulations] Publications: Press Coverage: 
The Mechanics of Curly Hair with: Jay Miller, Arnaud Lazarus and Basile Audoly 


In this study we seek to understand how natural curvature affects the configuration of a thin elastic rod suspended under its own weight. We combine precision desktop experiments, numerics, and theoretical analysis to explore the equilibrium shapes set by the coupled effects of elasticity, natural curvature, nonlinear geometry, and gravity. A phase diagram is constructed in terms of the control parameters of the system, namely the dimensionless curvature and weight, where we identify three distinct regions: planar curls, localized helices, and global helices. We analyze the stability of planar configurations, and describe the localization of helical patterns for long rods, near their free end. The observed shapes and their associated phase boundaries are then rationalized based on the underlying physical ingredients. Our framework is applicable to a variety of natural and engineered rodlike structures, over many length scales. Publications: Press Coverage: 
Geometrically nonlinear configurations of thin elastic rods with: Arnaud Lazarus and Jay Miller 


We have developed a novel continuation method to calculate the equilibria of elastic rods under large geometrically nonlinear displacements and rotations. To describe the kinematics we exploit the synthetic power and computational efficiency of quaternions. The energetics of bending, stretching and torsion are all taken into account to derive the equilibrium equations which we solve using an asymptotic numerical continuation method. This provides access to the full set of analytical equilibrium branches (stable and unstable), a.k.a bifurcation diagrams. This is in contrast with the individual solution points attained by classical energy minimization or predictorcorrector techniques. 
We challenge our numerics for the specific problem of an extremely twisted naturally curved rod and perform a detailed comparison against a precision desktopscale experiments. The quantification of the underlying 3D buckling instabilities and the characterization of the resulting complex configurations are in excellent agreement between numerics and experiments. 


We have also studied the buckling of a slender rod embedded in a soft elastomeric matrix. In our experiments, depending on the control parameters, both planar wavy (2D) or nonplanar coiled (3D) configurations are observed in the postbuckling regime. Our analytical and numerical results indicate that the rod buckles into 2D configurations when the compression forces associated to the two lowest critical modes are well separated. In contrast, 3D coiled configurations occur when the two buckling modes are triggered at onset, nearly simultaneously. We show that the separation between these two lowest critical forces can be controlled by tuning the ratio between the stiffness of the matrix and the bending stiffness of the rod, thereby allowing for specific buckling configurations to be target by design. Publications: 