# Coiling of elastic rods on rigid substrates

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Edited* by Harry L. Swinney, The University of Texas at Austin, Austin, TX, and approved September 2, 2014 (received for review May 16, 2014)

## Significance

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 paper, we combine precision model experiments with computer simulations of a rescaled analogue system 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.

## Abstract

We investigate the deployment of a thin elastic rod onto a rigid substrate and study the resulting coiling patterns. In our approach, we combine precision model experiments, scaling analyses, and computer simulations toward developing predictive understanding of the coiling process. Both cases of deposition onto static and moving substrates are considered. We construct phase diagrams for the possible coiling patterns and characterize them as a function of the geometric and material properties of the rod, as well as the height and relative speeds of deployment. The modes selected and their characteristic length scales are found to arise from a complex interplay between gravitational, bending, and twisting energies of the rod, coupled to the geometric nonlinearities intrinsic to the large deformations. We give particular emphasis to the first sinusoidal mode of instability, which we find to be consistent with a Hopf bifurcation, and analyze the meandering wavelength and amplitude. Throughout, we systematically vary natural curvature of the rod as a control parameter, which has a qualitative and quantitative effect on the pattern formation, above a critical value that we determine. The universality conferred by the prominent role of geometry in the deformation modes of the rod suggests using the gained understanding as design guidelines, in the original applications that motivated the study.

The laying of the first transatlantic telegraph cable (1) opened the path for fast long-distance communication. Nowadays, submarine fiber-optic cables, a crucial backbone of the international communications (e.g., the Internet) infrastructure, are typically installed from a cable-laying vessel that, as it sails, pays out the cable from a spool downward onto the seabed. The portion of suspended cable between the vessel and the contact point with the seabed takes the form of a catenary (2). Similar procedures can also be used to deploy pipelines (3), an historical example of which is the then highly classified Operation PLUTO (Pipe-Lines Under the Ocean) (4), which provided fuel supplies across the English Channel at the end of World War II. One of the major challenges in the laying process of these cables and pipelines is the accurate control between the translation speed of the ship, *v*. A mismatch between the two may lead to mechanical failure due to excessive tension (if

The common thread between these engineering systems is the geometry of deployment of the filamentary structure with a kinematic mismatch between the deposition rate and the translational speed. Moreover, the suspended catenary can be treated as a thin rod (9) given that the diameter of the cable, pipe, or filament can be orders of magnitude smaller than any other length scales in the system. The process of pattern formation for an elastic rod coiling on a substrate, also known as the elastic sewing machine (10), has been previously studied both numerically (11) and experimentally (10, 12). However, a systematic study and a predictive understanding of the underlying mechanisms that determine the coiling modes and set the length scale of the patterns remain remote. Moreover, there is a need for high-fidelity numerical tools that can capture the intricate geometric nonlinearities of the coiling process.

Here, we conduct a hybrid experimental and numerical investigation of the coiling of a thin elastic rod onto a moving substrate and characterize the resulting patterns. We perform precision experiments at the desktop scale (Fig. 1*A*), where a custom-fabricated rod is deposited onto a conveyor belt. As the relative difference between the speeds of the injector and the belt is varied, we observe a variety of oscillatory coiling patterns that include sinusoidal meanders (Fig. 1*B* and Movie S1), alternating loops (Fig. 1*C* and Movie S2), and translated coiling (Fig. 1*D* and Movie S3). Our model experiments explore the scale invariance of the geometric nonlinearities in the mechanics of thin elastic rods (9), thereby enabling a systematic exploration of parameter space. In parallel, we perform numerical simulations, using the discrete elastic rods (DER) method (9, 13, 14) that is introduced from computer graphics into the engineering community, and find good quantitative agreement with experiments.

Our investigation emphasizes (*i*) geometry, (*ii*) universality, and (*iii*) the significance of natural curvature. The patterns resulting from coiling of an elastic rod in our experiments have a striking resemblance to those found when deploying a viscous thread onto a moving belt (14⇓⇓⇓⇓–19) (known as the viscous sewing machine) and in electrospinning of polystyrene fibers (20). As such, (*i*) this similarity across various systems reinforces that geometry is at the heart of the observed phenomenon, whereas the constitutive description plays second fiddle. A fundamental challenge in this class of problems lies in the geometric nonlinearities that arise in the postbuckling regime, even though the material remains in the linear regime and small-strain elasticity is maintained (9). Furthermore, (*ii*) as we investigate the first mode of instability, from straight to meandering patterns, we observe that the onset is consistent with a Hopf bifurcation (21). Finally, (*iii*) we find that the natural curvature of the rod is a pivotal control parameter. This is important given that in the engineering systems mentioned above, natural curvature may develop from the spooling of the cables and pipes for storage and transport (22). Together, the experiments and numerics enable us to identify the physical ingredients and predictively understand the characteristic length scales that underlie the coiling process.

## Physical and Numerical Experiments

### Desktop-Scale Physical Experiments.

A photograph of our experimental apparatus is presented in Fig. 1*A*; an elastomeric rod is deployed at a controlled injection speed, *v*, onto a conveyor belt that is moving with speed *B1* and Movie S1), alternating loops (Fig. 1 *C1* and Movie S2), translated coiling (coiling only to one side, Fig. 1 *D1* and Movie S3), and stretched coiling (coils separated by a long catenary, see Fig. 3*B*). See *Materials and Methods* for additional details on the experiments.

### Numerics from the Graphics Community.

Hand in hand with the physical experiments, we conduct numerical simulations using the DER method (13, 14), which was originally developed to serve the visual special effects and animated feature film industries’ pursuit of visually dramatic (i.e., nonlinear, finite deformations) dynamics of hair, fur, and other rod-like structures. DER is based on discrete differential geometry (DDG), a budding field of mathematics that is particularly well suited for the formulation of robust, efficient, and geometrically nonlinear numerical treatments of elasticity (25). This method supports arbitrary (i.e., curved) undeformed configurations, arbitrary cross sections (i.e., noncircular), and dynamics. A direct comparison between simulations and experiments is provided in Movies S1–S3, with no fitting parameter; all control, geometric, and material parameters are measured independently.

## Physical Ingredients

We assume that due to its slenderness and the geometry of the setup, the rod is inextensible. The configuration of a Kirchhoff elastic rod (9, 26) is succinctly represented by an adapted framed curve

The energy stored in the deformation of the rod is expressed in terms of inertial, gravitational, and elastic contributions per unit length,*i*) elastic *ii*) gravitational *iii*) inertial *R* is the typical radius of curvature of deformation of the rod) that lies within

### Gravito-Bending Length.

We now identify the primary characteristic length scale of our system. We first consider the case of a planar, twist-free deformation *κ*. The corresponding bending energy is

## Static Coiling

We start our investigation by deploying the rod onto a steady substrate (belt speed *A* and *B*, *SI Appendix*, and Movies S4 and S5).

### The Role of Natural Curvature.

In Fig. 2*C*, we plot the dimensionless coiling radius as a function of the dimensionless natural radius of the rod,

Above, for a straight rod *C*, we extend the previous scaling analysis to consider a rod with natural curvature: When deformed to coil at radius *c* can be estimated by noting that, for a straight rod, the radius of curvature of its suspended portion at the contact point with the substrate (Fig. 2*C*, *Inset*) is of order *C*, *Inset* has radius *c* in Eq. **3** yields the solid line in Fig. 2*C*, which is in good agreement with both experiments and simulations.

### The Interplay Between Natural Curvature and Twist.

In Fig. 2 *A* and *B* we present snapshots of static coiling, using both a straight rod (Fig. 2*A*, *B*, *D*, we plot the normalized inversion length,

Given the good agreement between experiments and simulations found thus far, we now use the DER simulations to access quantities numerically that are challenging or impossible to obtain from the experiments. In Fig. 2*E* we plot simulated data for the time series (time is normalized by

To gain further insight into the nature of these two regimes we note that, by geometry, every deposited loop introduces a total twist of *SI Appendix*). Naturally curved rods prefer to twist along their heel; twisting along the deposited loop is costly in bending energy, due to the attendant misalignment between natural and actual curvature orientations. Rods with low or no natural curvature also begin by twisting along their heel (similar to their curved counterparts). Eventually, however, the accumulated twisting force along the heel overcomes the bending resistance along the deposited loop. As a result, twist begins to accumulate continuously in the deposited loop, rather than in the suspended heel (*SI Appendix*, Fig. S3).

### Coiling Inversion and the Critical Value of Twist.

We now seek to understand the finding that the coiling inversion occurs at a critical twist, **1**, we also need to include the twist energy per unit length, *SI Appendix*, Fig. S1*B*), and after a length *E*, leading to a coiling inversion.

The dependence of the inversion length on *D*) can now be understood as follows. If *N* loops are deposited between inversion events, this increases the mean dimensionless twist by *SI Appendix*), while consuming a rod segment of length *N* gives *D*.

## Dynamic Coiling

We proceed by investigating the dynamic coiling regime, where the rod is deployed at a controlled injection speed, *v*, onto a substrate that is now made to move, by switching on the conveyor belt in our apparatus with a speed, *v* and

### Phase Diagrams.

In Fig. 3*A* we present the phase diagram constructed from a systematic exploration of the *v* constant, we stepped up

We have also constructed the *SI Appendix*, Fig. S5*A*) and found that the meandering regime can be expanded significantly with increasing *SI Appendix*, where we also constructed the *SI Appendix*, Fig. S5*B*), finding that

### The Straight-to-Meandering Transition.

We now give special focus to the first mode of instability above *B* and Movie S1), where the rod prescribes a sinusoidal trajectory on the belt. In Fig. 4 *A* and *B* (for *C* and *D*, as a function of the control parameter *ε*, for two rods with *ε*, with a finite value at the onset of the instability,

We now investigate these dependences of the amplitude and wavelength of the sinusoidal meandering patterns, on the dimensionless speed mismatch, *ε*. We start by assuming that the rod is inextensible, such that the arc length of a single period, *l*, can then be related to its wavelength by **4** yields*SI Appendix*). Eq. **5** recovers the finite value *D*). In *SI Appendix*, the accuracy of this comparison is quantified further against the simulated data.

Our DER simulation tool also supports dynamics and can therefore capture transients caused by step variations of the control parameter, *ε*. These are, however, challenging to be systematically studied experimentally due to the excessive length of rod required. As such, and ensured by the excellent agreement between the experiments and numerics presented in Fig. 4 *C* and *D*, we use DER to quantify these transient dynamics. For example, in Fig. 4 *A* and *B*, the control parameter was instantaneously switched from the meandering states, *C*, we plot *ε*, before the switch to

Together, these observations on the meandering patterns combined—square root dependence of the amplitude on the control parameter, a finite onset wavelength, and critical slowing down at the onset of the instability—suggest that the meandering instability in our rod deployment pattern formation process is consistent with a Hopf bifurcation, which marks the transition from a stable to an oscillatory state in many other nonlinear systems. Moreover, it is interesting to note that the meandering instability for a viscous thread falling onto a moving belt has been shown to also arise through a Hopf bifurcation (17, 18), pointing to universality features and emphasizing the prominence of geometry in these two systems.

### Meandering Length Scales.

It remains to establish how the meandering amplitude and wavelength depend on the physical parameters of the problem: the gravito-bending length, *H*. For this purpose, we have performed a series of DER simulations in the meandering regime for rods with gravito-bending lengths in the range

In Fig. 4*F* we plot the values of dimensionless onset wavelength, *F*, *Inset* shows that *β* are numerical constants (derivation in *SI Appendix*), and using Eq. **5** we find**6** to describe the *F*, *Inset* for **6** yields **6** with Eqs. **4** and **5** allows us to predict the amplitude and wavelength, over arbitrary values of *ε* in the meandering regime, for rods with a wide range of mechanical properties and deployment heights. This prediction agrees with our observations from experiments and simulations, summarized in Fig. 4 *C*, *D*, and *F*.

### The Effect of Natural Curvature.

In the dynamic coiling regime *A* and *B*, we plot the normalized meandering length scales, *ε* can be calculated), as a function of

Whereas the amplitude and wavelength are left unmodified for lower values of **7**, we require *F*, in Fig. 5*C* we find that at a fixed dimensionless height *SI Appendix*). Even if *D* (DER simulations with **7** and **8**, along with the data in Fig. 5 *C* and *D*, allow us to estimate the critical natural curvature of rods over a wide range of physical parameters and deployment conditions.

## Conclusions

We have explored a rescaled analog system to study the geometrically nonlinear coiling during deployment of a thin elastic rod onto a substrate. We combined precision model experiments, computer simulations, and scaling analyses to predictively understand the physical parameters that determine the coiling patterns. We focused on static coiling; established the radius of coiling as a function of the gravito-bending length,

Our dimensionless formulation suggests that the problem is geometry dominated, conferring a universality of our findings across engineering applications of diverse spatial scales, from the microscopic (e.g., serpentine nanotubes) to the macroscopic (e.g., transoceanic cables and pipelines). Having generated phase diagrams for the control parameters, it becomes possible to target specific patterns. For example, meandering modes could be excited during the deployment of wires in a textile or pipelines onto the seabed, thereby conferring resilience under strain due to stretching of fabric or seismic activity, respectively. Because wires, cables, and pipelines are often manufactured, stored, and transported in spools that impart permanent curvature, our quantitative analysis could help predict the threshold spool radius beyond which these rodlike structures cannot be considered naturally straight. Understanding the participation of twist in static coiling and meandering could inform the design of application-specific rodlike structures, whose elastic response to twist and bending could be tuned separately; e.g., rotating joints could reduce twist effects, allowing small radii spools to deposit without static coiling inversion. Finally, considering additional practical ingredients such as fluid loading and complex topographies is a possible direction of future study that can now be readily tackled by further augmenting the framework that we have introduced.

## Materials and Methods

### Rapid Prototyping of Rod Samples.

The rods used in the experiments were cast with silicone-based rubber (vinylpolysiloxane, Elite Double 8 and 32; Zhermack), using PVC tubes as molds. To impart natural curvature to the rod, the tubes were first wrapped around cylindrical objects with the desired radii. The fluid mixture of polymer and catalyst was injected into each tube, which was carefully cut after the curing process to extract the soft elastic rod. For the fabrication of a straight rod, the mold was attached to a rigid straight bar. Two types of rods were used. The first had radius

### Experimental Setup.

The apparatus was composed of a conveyor belt with a vinyl surface and an injection system to deploy the rod, both of which were driven using stepper motors (MDrive). For consistency, we ensured that the rod was aligned at the injector

### Discrete Elastic Rods Simulation.

Our DER code represents the rod by a piecewise linear centerline, along with a per-segment material frame represented by its angular deviation from a reference frame (13, 14). See *SI Appendix* for more details. We implemented a simple contact model by applying Dirichlet boundary conditions (pinned nodes) at the points of rod–substrate contact. However, the edges (an edge connects two consecutive nodes) on the deposited rod that are within a certain arc length from the contact point can rotate about the rod centerline. We take this distance to be *SI Appendix*).

## Acknowledgments

We are grateful to Basile Audoly for enlightening discussions and thank J. Marthelot, J. Mannet, and A. Fargette for help with preliminary experiments. We acknowledge funding from the National Science Foundation (CMMI-1129894) and a donation of computer hardware by Intel Corporation.

## Footnotes

↵

^{1}M.K.J. and F.D. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: preis{at}mit.edu or eitan{at}cs.columbia.edu.

Author contributions: E.G. and P.M.R. designed research; M.K.J., F.D., J.J., E.G., and P.M.R. performed research; M.K.J., F.D., J.J., E.G., and P.M.R. contributed new reagents/analytic tools; M.K.J. and F.D. analyzed data; and M.K.J., F.D., E.G., and P.M.R. wrote the paper.

The authors declare no conflict of interest.

↵*This Direct Submission article had a prearranged editor.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1409118111/-/DCSupplemental.

## References

- ↵.
- Gordon J

- ↵.
- Carter L, et al.

*Submarine Cables and the Oceans: Connecting the World*, UNEP-WCMC Biodiversity Series No. 31 (ICPC/UNEP/UNEP-WCMC, Cambridge, UK) - ↵.
- Gerwick B

- ↵.
- Searle A

*PLUTO: Pipe-Line Under the Ocean*(Shanklin Chine, Shanklin, UK) - ↵
- ↵
- ↵
- ↵
- ↵.
- Audoly B,
- Pomeau Y

- ↵
- ↵.
- Mahadevan L,
- Keller J

- ↵
- ↵
- ↵.
- Bergou M,
- Audoly B,
- Vouga E,
- Wardetzky M,
- Grinspun E

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Cross M,
- Greenside H

- ↵
- ↵.
- Lazarus A,
- Miller J,
- Metlitz M,
- Reis PM

- ↵
- ↵.
- Bobenko A,
- Sullivan J,
- Schröder P,
- Ziegler G

*Discrete Differential Geometry*, Oberwolfach Seminars 38 (Birkhäuser, Basel) - ↵
- ↵
- ↵
- ↵
- ↵

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