Thin Rods

Understanding the mechanics of thin elastic rods under geometrically nonlinear configurations is a major thrust of our research. There is a recognized importance of filament-like 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 well-established 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.

Compressible and electrically conducting fibers for large-area sensing of pressures

with: Andreas Leber, Alexis Gérald Page, Dong Yan, Yunpeng Qu, Shahrzad Shadman, and Fabien Sorin

Flexible pressure sensors offer a wide application range in health monitoring and human–machine interaction. However, their implementation in functional textiles and wearable electronics is limited because existing devices are usually small, 0D elements, and pressure localization is only achieved through arrays of numerous sensors. Fiber‐based solutions are easier to integrate and electrically address, yet still suffer from limited performance and functionality. An asymmetric cross‐sectional design of compressible multimaterial fibers is demonstrated for the detection, quantification, and localization of kPa‐scale pressures over m2‐size surfaces. The scalable thermal drawing technique is employed to coprocess polymer composite electrodes within a soft thermoplastic elastomer support into long fibers with customizable architectures. Thanks to advanced mechanical analysis, the fiber microstructure can be tailored to respond in a predictable and reversible fashion to different pressure ranges and locations. The functionalization of large, flexible surfaces with the 1D sensors is demonstrated by measuring pressures on a gymnastic mat for the monitoring of body position, posture, and motion.

• A. Leber, A. G. Page, D. Yan, Y. Qu, Sh. Shadman Yazdi, P. M. Reis, and F. Sorin, “Compressible and electrically conducting fibers for large-area sensing of pressures” Adv. Funct. Mater., DOI: 10.1002/adfm.201904274 (2019).
[html, pdf].

Patterns of carbon nanotubes by flow-directed deposition on substrates with architectured topographies

with: M. K. Jawed, N. G. Hadjiconstantinou and D. M. Parks



We develop and perform continuum mechanics simulations of carbon nanotube (CNT) deployment directed by a combination of surface topography and rarefied gas flow. We employ the discrete elastic rods method to model the deposition of CNT as a slender elastic rod that evolves in time under two external forces, namely, van der Waals (vdW) and aerodynamic drag. Our results confirm that this self-assembly process is analogous to a previously studied macroscopic system, the “elastic sewing machine”, where an elastic rod deployed onto a moving substrate forms nonlinear patterns. In the case of CNTs, the complex patterns observed on the substrate, such as coils and serpentines, result from an intricate interplay between van der Waals attraction, rarefied aerodynamics, and elastic bending. We systematically sweep through the multidimensional parameter space to quantify the pattern morphology as a function of the relevant material, flow, and geometric parameters. Our findings are in good agreement with available experimental data. Scaling analysis involving the relevant forces helps rationalize our observations.

• M.K. Jawed, N. Hadjiconstantinou, D. Parks, P.M. Reis, “Patterns of carbon nanotubes by flow-directed deposition on substrates with architectured topographies” Nano Lett., 18, 1660-1667 (2018). [html, pdf]

Instabilities of a flexible helical rod rotating in a viscous fluid

with: Khalid Jawed



We combine experiments with simulations to investigate the fluid-structure interaction of a flexible helical rod rotating in a viscous fluid, under low Reynolds number conditions. Our analysis takes into account the coupling between the geometrically nonlinear behavior of the elastic rod with a nonlocal hydrodynamic model for the fluid loading. We quantify the resulting propulsive force, as well as the buckling instability of the originally helical filament that occurs above a critical rotation velocity. A scaling analysis is performed to rationalize the onset of this instability. A universal phase diagram is constructed to map out the region of successful propulsion and the corresponding boundary of stability is established. Comparing our results with data for flagellated bacteria suggests that this instability may be exploited in nature for physiological purposes.

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.

• M. K. Jawed, N. Khouri, F. Da, E. Grinspun and P.M. Reis, “Propulsion and instability of flexible helical rod rotating in viscous fluid” Phys. Rev. Lett. 115, 168101 (2015). [html, pdf]
• M.K. Jawed and P.M. Reis “Deformation of a soft helical filament in an axial flow at low Reynolds number” Soft Matter 12, 1898-1905 (2016). [html, pdf]
• M.K. Jawed and P.M. Reis, “Dynamics of a flexible helical filament rotating in a viscous fluid near a rigid boundary” Phys. Rev. Fluids 2, 034101 (2017). [html, pdf]

The interplay between the mechanics and topology of elastic knots

with: Khalid Jawed, Peter Dieleman and Basile Audoly



We combine experiments and theory to study the mechanics of overhand knots in slender elastic rods under tension. The equilibrium shape of the knot is governed by an interplay between topology, friction, and bending. We use precision model experiments to quantify the dependence of the mechanical response of the knot as a function of the geometry of the self-contacting region, and for different topologies as measured by their crossing number. An analytical model based on the nonlinear theory of thin elastic rods is then developed to describe how the physical and topological parameters of the knot set the tensile force required for equilibrium. Excellent agreement is found between theory and experiments for overhand knots over a wide range of crossing numbers.

• M.K. Jawed, P. Dieleman, B. Audoly and P.M. Reis “Untangling the Mechanics and Topology in the Frictional Response of Long Overhand Elastic Knots” Phys. Rev. Lett., 115, 118302 (2015). [html, pdf] (Supplemental Material [pdf]) (Physics Focus).

Press Coverage:
• Jennifer Chu “Untangling the mechanics of knots: New model predicts the force required to tie simple knots” MIT News, 8/9/15.
Spotlight on MIT’s Homepage. 9/9/15
• Michael Schirber, “Focus: Measuring the Forces in a Knot” Physics, 8, 86, 11/9/15.
• Edwin Cartlidge, “Physics may reveal how to tie the perfect knot” Science, 8/9/15.
• David Larousserie, “Un mystère dénoué: Des physiciens expliquent le lien entre la force d’un noeud et le nombre de tour dans la boucle” Le Monde, 14/9/15.
• Jennifer Ouellette, “What’s the Best Way to Tie Your Shoes? Physics May Have the Answer.” Gizmodo, 23/9/15.

Buckling of a rod inside a cylindrical constraint: Applications to coiled tubing operations

with: Jay Miller, Jahir Prabon and Nathan Wicks



We present results of an experimental investigation of a new mechanism for extending the reach of an elastic rod injected into a horizontal cylindrical constraint, prior to the onset of helical buckling. This is accomplished through distributed, vertical vibration of the constraint during injection. A model system is developed that allows us to quantify the critical loads and resulting length scales of the buckling configurations, while providing direct access to the buckling process through digital imaging. In the static case (no vibration), we vary the radial size of the cylindrical constraint and find that our experimental results are in good agreement with existing predictions on the critical injection force and length of injected rod for helical buckling. When vertical vibration is introduced, reach can be extended by up to a factor of four, when compared to the static case. The injection speed (below a critical value that we uncover), as well as the amplitude and frequency of vibration, are studied systematically and found to have an effect on the extent of improvement attained.

Image courtesy of Schlumberger-Doll Research.

• J.T. Miller, C.G. Mulcahy, J. Pabon, N. Wicks, P.M. Reis, “Extending the Reach of a Rod Injected into a Cylinder Through Distributed Vibration” J. App. Mech., 82, 021003 (2015) [html, pdf].
• J.T. Miller, T. Su, J. Pabon, N. Wicks, K. Bertoldi and P.M. Reis “Buckling of a thin rod inside a horizontal cylindrical constraint” Extreme Mechanics Letters, 3, 36-44 (2015) [html, pdf].
• J.T. Miller, T. Su, E.B. Dussan, J. Pabon, N. Wicks, K. Bertoldi and P.M. Reis “Buckling-induced lock-up of a slender rod injected into a horizontal cylinder”, Int. J. Solids Struct., 72, 153-164 (2015) [html, pdf].
• C. G. Mulcahy, T. Su, N. Wicks and P.M. Reis “Extending the Reach of a Rod Injected Into a Cylinder Through Axial Rotation” J. Appl. Mech., 83, 051003 (2016) [html, pdf].

Press Coverage:
• Jennifer Chu “Untangling how cables coil.” MIT News, 10/03/14.

Coiling ‘spaghetti’ onto rigid substrates

with: Eitan Grinspun



The deployment of a rodlike structure onto a moving substrate is commonly found in a variety engineering applications, from the fabrication of nanotube serpentines to the laying of submarine cables and pipelines. Predictively understanding the resulting coiling patterns is challenging given the nonlinear geometry of deposition.

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]

• M.K. Jawed, F. Da, J. Joo, E. Grinspun, and P.M. Reis “Coiling of elastic rods on rigid substrates” Proc. Natl. Acad. Sci. U.S.A., 111:41, 14663 (2014). [html, pdf]. (Supplementary Information, including videos [html, pdf]).
• M.K. Jawed, P.M. Reis “Pattern morphology in the elastic sewing machine” Extreme Mechanics Letters, 1, 76-82 (2015) [html, pdf]. (Supplementary Information [html, pdf]).
• M.K. Jawed, P.-T. Brun, and P.M. Reis “A geometric model for the coiling of an elastic rod deployed onto a moving substrate” J. App. Mech., 82(2), 1210007 (2015). [html, pdf]

Press Coverage:
• Jennifer Chu “Untangling how cables coil” MIT News, 10/03/14.
• Holly Evarts “How Things Coil” Columbia Engineering Press Release, 09/29/14.

The Mechanics of Curly Hair

with: Jay Miller, Arnaud Lazarus and Basile Audoly



We tackle the deceivingly simple problem of a suspended naturally curved rod, which we consider as an analogue for curly hair, to predict its resulting shapes. The role of natural curvature in the mechanics of rods, as a control parameter, has been largely overlooked in the literature.

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.

• J.T. Miller, A. Lazarus, Basile Audoly and P.M. Reis “Shapes of a Suspended Curly Hair” Phys. Rev. Lett, 112, 068103 (2014). [html, pdf] (Editor’s Suggestion & Cover Story).

Press Coverage:
• Denise Brehm “The physics of curly hair: Researchers develop first detailed model for a 3-D strand of curly hair.” MIT News, 02/13/14. [MIT homepage spotlight]; Carolyn Y. Johnson “Untangling the science of curly hair” Boston Globe, 02/14/14. Jeffrey Kluger “The Physics of Curly Hair—Because You Deserve to Know” TIME, 02/13/14. “Why Is Curly Hair Curly?” Slate, 02/14. Maggie Lange, “Wonderful New Research to Help Animators Create Better Curly Hair” New York Magazine, 02/24/14. Denise Chow “Curly Hair Physics Unraveled In New 3D Model Study” The Huffington Post, 03/03/14. Nicole Elphick, “The science of curly hair” Daily Life, 03/17/2014. Rachel Nuwer, “Graphic Geeks Can Now Give Their Characters Curly Locks” 05/01/2014.

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 predictor-corrector techniques.

We challenge our numerics for the specific problem of an extremely twisted naturally curved rod and perform a detailed comparison against a precision desktop-scale 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 non-planar coiled (3D) configurations are observed in the post-buckling 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.

• A. Lazarus, J.T. Miller and P.M. Reis “Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method” J. Mech. Phys. Solids., 61(8), 1712(2013) [html, pdf].
• A. Lazarus, J.T. Miller, M. Metlitz and P.M. Reis “Contorting a heavy and naturally curved elastic rod”, Soft Matter, 9 (34), 8274 (2013) [html, pdf]. (Special themed issue on “Geometry and Topology of Soft Materials”).
• T. Su, J. Liu, D. Terwagne, P.M. Reis and K. Bertoldi “Buckling of an elastic rod embedded on an elastomeric matrix: planar vs. non-planar configurations” Soft Matter, 10, 6294-6302 (2014). [html, pdf]. (Supplementary Information [html, pdf]).