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# iMechanica Journal Club (October 2013): Contortion of thin elastic rods

There has been a recent revival of the study of thin elastic rods from the mechanics, physics and computer science communities. From a fundamental perspective, predicticting the geometrically-nonlinear behavior of thin rods in the post-buckling regime is a challenging endeavor. Moreover, there are many modern industrial contexts for which rationalizing the mechanics of thin elastic filamentary structures is both relevant and timely. The role of intrinsic natural curvature of an elastic filament, which is often overlooked, is of particular interest since it can dramatically, quantitatively and qualitatively change the behavior of the system.

**1. The Mechanics of thin elastic rods - past and present:**

Thin elastic rods are long and slender structures that are nearly one-dimensional; their length is significantly larger than their diameter (Fig. 1). Instances of thin rods as structural elements appear in a wide range of contexts and length scales, both in the natural and built environment. Under different loading conditions, these structures can undergo mechanical instabilities that lead to large displacements and geometrically-nonlinear configurations that result in mechanical behavior that is challenging to rationalize and predict. The ‘ease’ and ‘softness’ in their reconfiguration provides outstanding kinematic freedom for function and practical applications. Examples include DNA [1], coiling of carbon nanotubes [2], bacteria flagella [3], conducting elements for stretchable electronics [4], filamentary plant structures [5], human hair [6], flexible cables and pipes [7] and coiled-tubing operations in the oil-gas industry [8].

**Figure 1.: Thin Elastic Rods. **a) Ensemble of thin elastic rods exhibiting geometrically-nonlinear behavior. b) Kinematics of a Cosserat rod in the global cartesian frame *(x,y,z)* The configuration of the rod is defined by its centerline, * r(s)*, as a function of the arclength,

*s*. The orientation of each mass point of the rod is represented by an orthonormal basis

*(*, called the directors, where

**d**1(s),**d**2(s),**d**3(s))*is constrained to be tangent to*

**d**3(s)*[6].*

**r**(s)The analysis of the mechanics of elastic rods has a long and distinguished history, a thorough description of which is beyond the scope of this post. In short, the earlier studies on the mechanics of rods are often traced back to the experimental buckling investigations of Musschenbroek in 1721 [6] and the analytic work of Euler in 1744 [9] for planar deformations of straight rods. Kirchhoff later derived the general equations for elastic rods in 1859 [10]. Kirchhoff’s kinetic analogy identified a correspondence between the equilibria of a rod and the motion of a spinning top [11]. This elegant connection allowed for a deeper understanding of mechanical instabilities and nonlinear equilibrium configurations of rods with natural curvature [12].

Many people, especially since the 1980’s, have been making important contributions in this area. A far-from-exhaustive list of researchers who have been making substantial contributions to the field is given at the end of this post. For a more detailed account of the mechanics of elastic rods, including a review of the modern state-of-the-art of the theory, we suggest the recent book by Basile Audoly and Yves Pomeau - “Elasticity and Geometry: From Hair Curls to the Non-linear response of Shells” (OUP) [6].

**2. Our take on thin rods:**

In my research group we have recently revisited the classic field of the mechanics of thin elastic rods and have been developing a research program that combines experimental, numerical and theoretical efforts. Surely, a lot has been done by previous giants in the field but we have also identified a number of new opportunities. One specific direction that we are currently pursuing involves developing tools for better understanding the non-trivial role of intrinsic natural curvature on thin rods. This is particularly relevant to many engineering problems where fibers, filaments, cables and pipes are spooled for storage and transport, which may irreversibly impart a natural curvature that must be treated as an independent variable. As others before, we have found that intrinsic natural curvature can lead to nontrivial and counter-intuitive effects that can dramatically change, both qualitatively and quantitatively, the behavior of the system, thereby calling for a thorough predictive physical understanding.

**2.1. Precision model experiments:**

During the deformation process of a thin rod, the large displacements involved can lead to non-negligible geometric nonlinearities, even if the material properties remain linear, that are mostly system independent. These universal modes of deformation underlie the fact that similar phenomena can be observed over a wide range of lengthscales (small and large). This opens the way to approach application in a unified way, without the need to work at the original scale that motivated the problem. In this context, we have been developing an experimental methodology for studying the mechanics of geometrically-nonlinear scenarios that has *Precision Model Experiments* (desktop-scale) at its basis. These model experiments allow for a reduction of the problem to its bare essential physical ingredients, that take advantage of the geometrically-rooted behavior of thin rods. Given the often non-trivial and counter-intuitive behavior of these systems, in addition to their use for model/numerics validation purposes, these model experiments also play an invaluable role as a tool for discovery and exploration.

**Figure 2.: Experiments with thin elastic rods.** a,b) Fabrication of our thin rods with custom natural curvature, *ko*, through casting of an elastomere inside a PVC flexible tube wound around a rigid cylinder of set diameter. c) Collection of elastic rods with different natural curvatures, *ko*. d) The writhing experiment. A naturally curved elastomeric rod is clamped to two concentrically aligned drill chucks, which allow for the end-to-end displacement or rotation of the extremities to be imposed.

Using rapid prototyping techniques, we cast our own rods with full custom control of material and geometric properties (Fig. 2). A flexible PVC tube, which acts as a mold (inner and outer diameter, *Di=3.1mm* and *Do=5mm*, respectively) is first wound around a cylinder of external radius *Re* (see Fig. 2a,b). Vinylpolysiloxane (VPS), an elastomer, is then injected into the PVC tube, which eventually cross-links at room temperature. After a setting period (typically of 24 hours) the outer flexible PVC pipe is cut to release the inner slender VPS elastic rod with constant natural curvature *ko=1/(Re+Do/2)*. Using cylinders with different radius *Re*, we can systematically produce rods with different custom values of natural curvature, typically in the range 0<*ko*<60m-1, which allow *ko* to be treat it as a control parameter that can be varied systematically. Whereas we also have control over the other geometric and material parameters, we typically fix them at: Young’s modulus, *E~0.3-1.3MPa*, density *ρ=1200 kg/m-3*, Poisson’s ratio *ν~0.5* and rod’s radius *R~1mm*. In Fig. 2c we show a collection of rods with different values of intrinsic natural curvature *ko*.

**2.2. Path-following numerical tools:**

In parallel to the physical experiments we have developed our own simulation tool that enables us to obtain the equilibrium configurations, as well as compute the full bifurcation diagrams of the system, which may include multiple stable and unstable states. We used MANlab [13], a continuation software package that we recently adapted for the specific use with elastic rods [14, 15]. The 3D kinematics of the rod is treated in a geometrically exact way by parameterizing the position of the centerline and making use of quaternions to represent the orientation of the material frame. The rod is assumed to be inextensible and unshearable. The equilibrium equations and the stability of their solutions are derived from the mechanical energy which takes into account the contributions due to internal moments (bending and twist), external forces and torques. Our use of quaternions allows for the equilibrium equations to be written in a quadratic form and solved efficiently with an asymptotic numerical continuation method. This finite element perturbation technique gives interactive access to semi-analytical equilibrium branches, in contrast with the individual solution points obtained from classical minimization or predictor–corrector techniques. Our simulation tool based on MANlab has similar capabilities to the path-following software AUTO [16] but with a number of added advantages including interactivity, user-friendliness and efficiency. More details about our numerical method can be round in the following reference [14]:

**3. The writhing of a thin rod as a test-bed:**

As an example of our approach, we test-ride our experimental and numerical tools by tackling the classic problem of writhing of a thin rod (which has received much studies in the past [17]) and has become a canonical model-system in which to investigate the equilibrium configurations, stability and spatial localization of contorted filamentary structures. In Fig. 2d we show a photograph of our experimental set-up. Upon fabricating our own rods as described above, the rod sample is then suspended between two horizontal concentric chucks which are used to apply an end-to-end rotation to the rod, while fixing the distance between the clamps. We have also studied the converse scenario of varying the end-to-end displacement of the clamps without rotation. We highlight that the primary control parameter in this study is the intrinsic natural curvature of the rod, which we vary systematically. More details on this study can be found in [15]:

A variety of experimental and numerical configurations for rods with increasing end-to-end rotation are presented in Fig. 3. Fig. 3a presents configurations for a naturally straight rod (*ko=0 m-1*) and Fig. 3b presents configuration for a naturally straight rod (*ko=44.8 m-1*), with the two cases being strikingly different from both qualitative and quantitative viewpoints.

**Figure 3.: The writhing experiment. **Top views of experimental and numerical equilibrium conﬁgurations for increasing values of the end-to-end rotation angle. The experimental pictures have a black background and the simulations have a white background. The simulation results are rendered to visualize twist by using bi-color rods. a) *ko=0 m-1* b) *ko=44.8 m-1*.

In the video below we present the corresponding comparison between experimental and numerical configurations, as a function of end-to-end rotation. Each of the four panels in the video corresponds to rods with increasing natural curvature and the video stops as soon as a plectoneme (localized deformation) forms. Again, we find excellent quantitative and qualitative agreement between the two, with no fitting parameters. Note that we also take the effect of self-weight into account in the simulations.

We have uncovered the original effect that weight delays the effect of natural curvature. Below a critical value of k_o^{crit}, gravity balances the imposed geometry and the heavy rod can be considered as being naturally straight, albeit with a small imperfection k_o. In contrast, above k_o^{crit}, the effect of natural curvature is significant and sufficient to break the symmetry of the rod’s pattern formation. We propose that this critical curvature is set by the inverse of the elastro-gravity lengthscale, *1/L_gb=[EI/(ρg S)]^{-1/3}*, where *EI* is the bending modulus of the rod, *ρS* is its linear mass density and *g* is the gravitational acceleration. Counterintuitively, we also find that imparting a constant natural curvature to our rods (essentially adding a geometric imperfection to the stress-free configuration) results in considerably postponing (in our particular study by approximately 43%) the emergence of the plectoneme instability, which is often synonymous of failure in practical systems.
This study highlights the power of combining our precision model experiments, which are used primarily as a tool of discovery in addition to validation, with our own simulation tools, which are then used to help rationalize the process by enabling access to quantities unavailable experimentally. More details can be found in [15].

**4. Ongoing work:**

Geared with both our precision model experiment approach and our simulation toolbox, we are now actively pursuing a variety of other problems involving geometrically-nonlinear configuration of thin rods in a variety of configurations, some of which with direct industrial relevance (and sponsorship). In collaboration with Eitan Grinpun’s group (Columbia University), we are also exploring the porting of a simulation tool - Discrete Elastic Rod Method [18] - that was originally developed in the context of physically-based computer animation, into engineering as a predictive tool. We hope to share these latest developments with the community in the near future.

**5. Concluding thoughts:**

In summary, we have identified new opportunities in reviving the study of the mechanics of thin rods, with a focus on geometrically-nonlinear behavior in the far-from-threshold post-buckling regime. In addition to a mechanism for validation, precision model experiments can play an important role as a tool for exploration and discovery in these nonlinear and often counter-intuitive processes. The role of intrinsic natural curvature of an elastic filament, which is often overlooked, is of particular interest since it can dramatically, quantitatively and qualitatively change the behavior of the system. Natural curvature arises naturally in many instances of filamentary structures during their fabrication, storage and transport in spools. These problems in naturally curved elastic rods pose significant challenges from a fundamental perspective, while having direct relevance and impact on practical applications.

**Questions for discussion:**

I would like to finish by opening the “i-floor” for discussion by posing a few questions:

- What opportunities (if any) do you see in reviving the classic subject of the mechanics of thin rods in modern contexts? ‘Is it all done’ or do you envision exciting open research directions?
- What modern tools (experimental, numerical and theoretical) do you use to approach this class of problems?
- Have you come across challenges in describing the geometrically-nonlinear behavior of thin rods/filaments in your own domain?
- Are you currently working on problems involving the mechanics of thin elastic filaments? If so, could you please share your recent work?

**Acknowledgments:** I would like to thank the members of my research group, the EGS.Lab at MIT, for all their hard work and creativity, especially, Arnaud Lazarus, James Miller, Tianxiang Su and Khalid Jawed who have worked on some of the material mentioned in this post. I am also grateful to my colleagues Eitan Grinspun, Basile Audoly, Katia Bertoldi and Nathan Wicks for fruitful collaborations on the mechanics of thin rods. Finally, I am grateful to NSF (CMMI-1129894), Schlumberger and Saint-Gobain for financial support.

**References most directly related to this post:**

For more information on research on the mechanics of slender structures from our research group - EGS.Lab: Elasticity, Geometry and Statistics Laboratory (MIT) - please visit: http://web.mit.edu/preis/www/

**Active Mechanicians working on thin rods:**

Here is a list of researchers who have been making substantial contributions to the field of the mechanics of thin elastic rods (in alphabetical order):

- Basile Audoly (CNRS/UPMC),
- Alan Champneys (Bristol),
- Bernard Coleman (Rutgers),
- Raymond Goldstein (Cambridge),
- Alain Goriely, (Oxford),
- Eitan Grinspun, (Columbia)
- Tim Healey (Cornell),
- Gert van der Heijden (UCL),
- Kathleen Hoffmann (Maryland)
- John Maddocks (EPFL),
- L. Mahadevan (Harvard),
- Rob Manning, (Haverford),
- Sébastien Neukirch (CNRS/UPMC),
- Oliver O'Reilly (UC Berkeley)
- Michael P Paidoussis (McGill),
- Prashant Purohit (UPenn),
- David Swigon, (Pittsburgh)
- Michael Tabor (Arizona),
- J. Michael Thompson (UCL).

Hopefully your contributions and replies below to this post will help further complement and increase this far-from-exhaustive list. Feel free to continue to send suggestions to add to this list.

**References:**

[1] S. Neukirch, Phys. Rev. Lett., **93**, 198107 (2004).

[2] N. Geblinger, A. Ismach and E. Joselevich, Nature Nanotech., **3**, 195 (2008).

[3] K. Son, J.S. Guasto and R, Stocker, Nature Physics **9**, 494 (2013).

[4] Y. Sun, W.M. Choi, H. Jiang, Y.Y. Huang, and J.A. Rogers, Nature Nanotech. **1**(3) 201 (2006).

[5] A. Goriely, M. Tabor. Phys. Rev. Lett. **80**(7) 1564 (1998).

[6] B. Audoly and Y. Pomeau. “Elasticity and geometry: from hair curls to the non-linear response of shells.” Oxford: Oxford University Press (2010).

[7] S. Goyal, N. Perkins and C. Lee, Int. J. Nonlinear Mech., **43**, 65 (2008).

[8] N. Wicks, B.L. Wardle and D. Pafitis. Int. J. of Mech. Sci. **50**(3) 538 (2008).

[9] R. Levien, "The Elastica: A Mathematical History". UCB/EECS-2008-103, EECS Department, University of California, Berkeley.

[10] E.H. Dill, Arch. Hist. Exact Sci. **44**(1) 1 (1992).

[11] M. Davies and F. Moon, Chaos, **3**, 93 (1993).

[12] A. Goriely and M. Tabor, Physica D 105, **20** (1997); A. Champneys, G. Van der Heijden, and J. Thompson, Philos. Trans. R. Soc. London, Ser. A **355**, 2151 (1997); P. Furrer, R. Manning, and J. Maddocks, Biophys. J. **79**, 116 (2000).

[13] S. Karkar, R. Arquier, and B. Cochelin, User Guide MANLAB 2.0 (2011).

[14] A. Lazarus, J. Miller, and P.M. Reis, J. Mech. Phys. Solids **61**, 1712 (2013).

[15] A. Lazarus, J. Miller, M. Metlitz, and P.M. Reis, Soft Matter **9** (34), 8274 (2013).

[16] E.J. Doedel, "AUTO: A program for the automatic bifurcation analysis of autonomous systems." Congr. Numer. **30** 265-284 (1981).

[17] A. Greenhill, Proc. – Inst. Mech. Eng., **34**, 182 (1883). A. Love, "A treatise on the mathematical theory of elasticity", University Press, (1920). J. Coyne, IEEE J. Oceanic Eng.,**15**, 72 (1990). J. Thompson and A. Champneys, Proc. R. Soc. London, Ser. A, **452**, 117 (1996). A. Goriely and M. Tabor, Proc. R. Soc. London, Ser. A, **454**, 3183 (1998). A. Goriely and M. Tabor, Proc. R. Soc. London, Ser. A, **453**, 2583 (1997).

[18] M. Bergou, M. Wardetzky, S. Robinson, B. Audoly and E. Grinspun "Discrete elastic rods." ACM Transactions on Graphics (TOG). **27**(3) 63 (2008).

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## Comments

## Thanks for bringing this interesting topic

Dear Pedro,

Thank you very much for bringing this interesting topic to the discussion. Although I have never worked on this field by myself, I am deeply attracted by the elegance of the elastic thin rod models developed by all of you.

Thin rods can be ubiquitously found in biological systems with different sizes, such as DNA double helix, microtubule, actin filament ,intermediate filament,various hairy structures and so on. Some of those are indeed intrinsically curved. Using elastic rod models to interpret relevant biological phenomena can be exciting. Of course, additional effects need to be added into the models. For example, thermal fluctuations can play an important role.

I believe elastic rod models have been used to explain diverse phenomena. Pedro, could you please list some examples of these?

Thanks

Shengqiang

## Electrostatic and thermal effects at small scales

Hi Shengqiang,

Absolutely. In the original text above I focused primarily on thin elastic rods where we also take self-weight (gravity) into account. This is relevant in the context of large scale applications.

However, the Asymptotic Numerical Method that we implemented in MANlab with my former post-doc, Arnaud Lazarus (now Assistant Professor at UPMC, Paris) is general and could be made to incorporate other mechanical ingredients. This include internal stretching, hydrostatic loading and contact forces arising, for instance, due to self-contact or when the rod interacts with external boundaries.

You mention, in particular, applications at small scale. Indeed, here i) electrostatic interactions and ii) thermal effects must be taken into account.

E.g. in the context of the formation of plectonemes (localized solutions) in DNA, I suggest the following articles:

Clauvelin, Nicolas, Basile Audoly, and S. Neukirch. "Elasticity and electrostatics of plectonemic DNA." Biophysical journal 96.9 (2009): 3716-3723.

N. Clauvelin, B. Audoly, and S. Neukirch. "Mechanical response of plectonemic DNA: An analytical solution" Macromolecules 41(12):4479-4483, 2008

Vologodskii, A.; Cozzarelli, N. "Modeling of long-range electrostatic interactions in DNA". Biopolymers 1995, 35 (3), 289–296.

Including the role of thermal effects at small scales is, as you point out, also very important. This requires a more statistical mechanical approach. Many people are working in this directions right now. As an example, I suggest:

Tianxiang Su, Somes K. Das, Ming Xiao, Prashant K. Purohit, “Transition between two regimes describing internal fluctuation of DNA in a nanochannel” PLoS One Vol. 6, e16890, (2011).

Perhaps others can chime in highlighting more relevant papers in the general area of electrostatics and temperature-driven statistical mechanics effects? Also, can people suggest other examples at small scales that employ rod-like frameworks?

Best,

Pedro

## Elastic rod

Thank you Pedro for posting this theme and discusssion. It is very well written and I enjoy reading it.

I would like to share some of my feelings working on the mechanics of rods in the previous few years. I saw many researchers using the Kirchhoff rod model to tackle interesting problems on rod mechanics. I learned this model from Prof. John Bassani when I was at Penn. However, when I tried to use the Kirchhoff model in my own research a few years ago, I found a lot of formulae in the literature are for

unshearable and inextensiblerods. I was trying to look for a version that takes into account the shear and extension because I needed those ingredients in my problem, but it was a pain for me, it was just difficult to find one that I can easily understand.As I learned from Prof. Bassani and my advisor Prof. Purohit at Penn, when a rod is extensible, we need to distinguish reference arc length and deformed arc length, and we need to be very careful when computing the strains because the differentiation should be done with respect to the reference arc length instead of the deformed one. I made mistakes for a few times before on this issues and arrived at wrong conclusions. I think this is one of the things that make a general Kirchhoff rod more complicated.

After a while I got this paper that I found very useful for me:

Yaoming Shi, Andrey E. Borovik, John E. Hearst.

Elastic rod model incorporating shear and extension, generalized nonlinear Schrodinger equations, and novel closed-formed solutions for supercoiled DNA.J. Chem. Phys. 1995.The most useful thing for me is not the DNA problem (it is interesting though), but that the rod model formulae there are pretty clearly written and relatively easy for me to understand and incorporate into my own simulation code. The formulae there may be cited from other older papers but I am not sure.

I think it would be great if some one knows a review paper that summarizes different rod models and their assumptions. I think that will be helpful for the students working in this field. There are different frames, frenet frame, materails frame, twistless bishop frame, it took me a while to understand these from different places.

At the end, I would like to share my recent work with Katia, Nathan and Jahir. We tried to understand how a confined rod buckles in the present of friction. We concluded that with friction, a rod can be stable even when the determinant of the stiffness matrix is negative. The critical load will be a function of the system imperfection. I think it is interesting and welcome any comments.

Su T, Wicks N, Pabon J, Bertoldi K. Mechanism by which a frictionally confined rod loses stability under initial velocity and position perturbations. International Journal of Solids and Structures. 2013;50:2468-2476

Thanks.

Sincerely,

Tianxiang Su

## More theoretical papers that address natural curvature of rods

I would like to share with you a few more papers, of which I just became aware, on the theory of the mechanics of thin rods that also takes natural curvature into account in a variety of contexts.

Adhesion:

http://www.sciencedirect.com/science/article/pii/S0093641313000177

Encirclement:

http://prl.aps.org/abstract/PRL/v99/i15/e158103

Tree-like structures

http://rspa.royalsocietypublishing.org/content/468/2137/206

Plant-growth:

http://www.sciencedirect.com/science/article/pii/S0020768310004427

Many thanks to Oliver O'Reilly (University of California at Berkeley) for bringing these to our attention.

Pedro

## Thermal motions of rods

Thanks, Pedro for an interesting post. The field of rod mechanics is quite mature, but that does not mean there are not interesting problems to solve. There is some very nice work on knots and yarns that relies on mechanics of rods. The mechanics of thin elastic strips and how it can be related to plate theories is also very interesting. I have been primarily interested in the application of rod mechanics to DNA and other biofilaments. A key ingredient that is needed to do this to understand the effect of thermal fluctuations. In some cases electrostatics in solution is also a very important ingredient because DNA and other biofilaments are charged objects. In most of these problems we need to account for geometrically non-linear effects. Here are a few papers in which my students and I have done this:

1. David Argudo and Prashant K. Purohit. "The dependence of DNA supercoiling on solution electrostatics", Acta Biomaterialia 8, 2133-2143, (2012).

2. David Argudo and Prashant K. Purohit, "Competition between DNA supecoils and toroids in single molecule DNA condensation", Biophysical Journal 103, 118-128, (2012).

3. Tianxiang Su and Prashant K. Purohit, "Thermomechanics of a heterogeneous fluctuating chain", J. Mech. Phys. Solids 58, 164-186, (2010).

Of course, thermal motion of filaments is relevant only at micro/nano scales. But, there are really interesting problems in rod and shell mechanics at the macroscopic scales with applications in plant biology. People had realized this a while ago but each year we continue to see creative new applications.

Prashant Purohit.

## Very Interesting Work!

Hi Pedro,

Thanks for writing this beautiful post! I admittedly know very little about the topic but a couple of questions came to mind when I read your post. First, in this post, you seem to have discussed systems which have linear stress-strain relations (the nonlinearity is instead due to geometric nonlinearities). Are there interesting observations for thin inelastic rods that have both geometric nonlinearity and material nonlinearity? For instance, what about strain-hardening, elastic-plastic, and strain-softening materials? Are there applications where the material nonlinearity can be exploited? As a related question, how do defects affect these thin elastic rods? For instance, if you intentionally introduce a thinned region or a notch into your rods, do you see anything interesting?

Thanks so much,

Matt Pharr

## Nonlinear v.s. Geometric nonlinearities on thin rods

Hi Matt,

You make make some excellent points on the role of material nonlinearities in the class of slender systems that we are studying. By construction, our model experiments focus primarily on elastic materials (e.g. we cast our rods with elastomers to simplify these issues) in order to decouple the study of geometric nonlinear phenomena from nonlinear constitutive effects, with a focus on the former.

As you point out, engineering-relevant rod-like systems (pipes, cables, fibers, etc) are indeed typically made out of materials that require a nonlinear constitutive descriptions. If these systems are sufficiently slender, however, the axial strains and curvatures tend to be low enough (well below 1%) that the material remains mostly on the linear elastic regime, even in strongly geometric nonlinear configurations. In our work, we have chosen to focus on predictively understanding precisely these geometric nonlinearities (knowing too well that if we continue to drive/load the system, our material description may eventually no longer be appropriate).

Let me give you a couple of examples.

Sometime ago John Lister (Cambridge) and later Stephen Morris (Toronto) reported on the fascinating problem of a viscous thread falling on a moving belt which generates some fascinating patterns. You can find a representative video on the following link:

http://www.youtube.com/watch?v=CMYISqxS3K4

We have been working on the elastic version of this problem where we deploy a thin elastic rod also onto a moving belt and find nonlinear patterns that have a direct counterpart with the viscous case. Here the difference in constitutive behavior is dramatic: linear stress vs. strain-rate for the fluid case (surface tension must also be taken into account) when compared to linear stress-strain for our elastic case. The reason why we find such similar patterns is directly related to the strong geometric rooting of the problem. We hope to be able to report more detailed results in the near future.

As a second example, we are also working with an industrial partner on understanding large reconfigurations of coil-tubing (steel piping used in the oil & gas industry). Surely steel exhibits many of the material behavior you highlight but we have found that a Kirchhoff-like model with a linear elastic material to be extremely useful in allowing us to rationalize a series of non-trivial buckling instabilities that happen, well before the pipe undergoes irreversible plastic behavior (which we are trying to avoid in this particular case).

In any case, I am positive that there are many other problems where, even through the rod or filament is slender, high strains/curvatures may be attained for which a more sophisticated material description will be required.

In terms of constructively exploiting material nonlinearities on this class of systems, none come to mind, but if you have some ideas, I am all hears!

I would love to hear from others colleagues on their experiences with slender rod-like systems where they have had to deal with nonlinear-material issues.

Best,

Pedro

## Disorder

Thanks for a very interesting read! I'm interested in analogous phenomena in the energy functionals arising in condensed matter e.g. for 1D superfluids.

I was wondering if you know of any work where the integrability of the dynamic analog (rigid body) is broken by the presence of disorder i.e. randomness in the elastic properties. Since the rod can be mapped to the nonlinear Schrodinger equation, this should be connected with the problem of Anderson localization in condensed matter.

Best,

Austen

## Dear Austen, Sounds like

Dear Austen,

Sounds like an interesting question/observation but, to be honest, I really do not know how to answer. Perhaps other collagues may be able to address this?

Best,

Pedro