iMechanica - geometric nonlinearities
https://imechanica.org/taxonomy/term/9195
enRising Beyond Elastocapillarity
https://imechanica.org/node/19932
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1101">swelling</a></div><div class="field-item odd"><a href="/taxonomy/term/464">bending</a></div><div class="field-item even"><a href="/taxonomy/term/11087">elastocapillary</a></div><div class="field-item odd"><a href="/taxonomy/term/667">beams</a></div><div class="field-item even"><a href="/taxonomy/term/9195">geometric nonlinearities</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Douglas P. Holmes, P.-T. Brun, Anupam Pandey, and Suzie Protière, <em>Soft Matter</em>, <strong>12</strong>, 4886-4890, (2016).</p>
<p>We consider the elastocapillary rise between swellable structures using a favorable solvent. We characterize the dynamic deformations and resulting equilibrium configurations for various beams. Our analysis reveals the importance of the spacing between the two beams, and the elastocapillary length, which prescribes the relative magnitude of surface tension and bending stiffness in the system. In particular, we rationalize the transition between coalescence-dominated, bending-dominated, and swelling-dominated regimes, and enumerate the subtle interfacial mechanisms at play in the ratcheting of a fluid droplet trapped between the curling beams.</p>
<p> </p>
<p><img src="http://www.bu.edu/moss/files/2016/06/elastocap-swell.gif" alt="" width="702" height="868" /></p>
<p>Link: <a href="http://pubs.rsc.org/en/content/articlelanding/2016/sm/c6sm00329j#!divAbstract">http://pubs.rsc.org/en/content/articlelanding/2016/sm/c6sm00329j#!divAbstract</a></p>
<p> </p>
</div></div></div>Thu, 02 Jun 2016 13:36:00 +0000Douglas P Holmes19932 at https://imechanica.orghttps://imechanica.org/node/19932#commentshttps://imechanica.org/crss/node/19932iMechanica Journal Club (October 2013): Contortion of thin elastic rods
https://imechanica.org/node/15442
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/347">elasticity</a></div><div class="field-item odd"><a href="/taxonomy/term/855">journal club</a></div><div class="field-item even"><a href="/taxonomy/term/1636">experiments</a></div><div class="field-item odd"><a href="/taxonomy/term/2099">Geometry</a></div><div class="field-item even"><a href="/taxonomy/term/9194">Thin elastic rods</a></div><div class="field-item odd"><a href="/taxonomy/term/9195">geometric nonlinearities</a></div><div class="field-item even"><a href="/taxonomy/term/9196">mechanical instabilities</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
</p>
<p>
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.
</p>
<p>
<strong>1. The Mechanics of thin elastic rods - past and present:</strong>
</p>
<p align="justify">
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].
</p>
<p>
</p>
<p> <img src="http://web.mit.edu/preis/www/images/iMechanica_Oct2013/PedroReis_iMechanica_Fig1.jpg" alt="iMechanica Journal Club October 2013 | Pedro Reis | MIT" title="Contorting Thin Elastic Rods | Fig. 1" hspace="10" vspace="10" width="500" height="329" /></p>
<p>
</p>
<p>
<strong>Figure 1.: Thin Elastic Rods. </strong>a) Ensemble of thin elastic rods exhibiting geometrically-nonlinear behavior. b) Kinematics of a Cosserat rod in the global cartesian frame <em>(x,y,z)</em> The configuration of the rod is defined by its centerline, <em><strong>r</strong>(s)</em>, as a function of the arclength, <em>s</em>. The orientation of each mass point of the rod is represented by an orthonormal basis <em>(<strong>d</strong>1(s), <strong>d</strong>2(s),<strong>d</strong>3(s))</em>, called the directors, where <em><strong>d</strong>3(s)</em> is constrained to be tangent to <em><strong>r</strong>(s)</em> [6].
</p>
<p align="justify">
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].
</p>
<p align="justify">
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].
</p>
<p><strong>2. Our take on thin rods:</strong></p>
<p align="justify">
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.
</p>
<p align="justify">
<strong>2.1. Precision model experiments:</strong>
</p>
<p align="justify">
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 <em>Precision Model Experiments</em> (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.
</p>
<p> <img src="http://web.mit.edu/preis/www/images/iMechanica_Oct2013/PedroReis_iMechanica_Fig2.jpg" alt="iMechanica Journal Club October 2013 | Pedro Reis | MIT" title="Contorting Thin Elastic Rods | Fig. 2" hspace="10" vspace="10" width="700" height="240" /></p>
<p>
<strong>Figure 2.: Experiments with thin elastic rods.</strong> a,b) Fabrication of our thin rods with custom natural curvature, <em>ko</em>, 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, <em>ko</em>. 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.
</p>
<p align="justify">
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, <em>Di=3.1mm</em> and <em>Do=5mm</em>, respectively) is first wound around a cylinder of external radius <em>Re</em> (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 <em>ko=1/(Re+Do/2)</em>. Using cylinders with different radius <em>Re</em>, we can systematically produce rods with different custom values of natural curvature, typically in the range 0<<em>ko</em><60m-1, which allow <em>ko</em> 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, <em>E~0.3-1.3MPa</em>, density <em>ρ=1200 kg/m-3</em>, Poisson’s ratio <em>ν~0.5</em> and rod’s radius <em>R~1mm</em>. In Fig. 2c we show a collection of rods with different values of intrinsic natural curvature <em>ko</em>.
</p>
<p align="justify">
<strong>2.2. Path-following numerical tools:</strong>
</p>
<p align="justify">
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]:
</p>
<p>
<a href="http://web.mit.edu/preis/www/mypapers/reis_JMPS_ContinuationRods_2013.pdf" title="Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method">A.Lazarus, J.T. Miller and P.M. Reis "Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method"<br />
J. Mech. Phys. Solids., 61(8), 1712 (2013). </a>
</p>
<p>
</p>
<p><strong>3. The writhing of a thin rod as a test-bed:<br />
</strong></p>
<p align="justify">
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]:
</p>
<p>
<a href="http://web.mit.edu/preis/www/mypapers/lazarus_reis_SoftMatter_ContorningRods.pdf" title="Contorting a heavy and naturally curved elastic rod">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). </a>
</p>
<p align="justify">
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 (<em>ko=0 m-1</em>) and Fig. 3b presents configuration for a naturally straight rod (<em>ko=44.8 m-1</em>), with the two cases being strikingly different from both qualitative and quantitative viewpoints.
</p>
<p>
</p>
<p> <img src="http://web.mit.edu/preis/www/images/iMechanica_Oct2013/PedroReis_iMechanica_Fig3.jpg" alt="iMechanica Journal Club October 2013 | Pedro Reis | MIT" title="Contorting Thin Elastic Rods | Fig. 3" hspace="10" vspace="10" width="700" height="231" /></p>
<p>
</p>
<p>
<strong>Figure 3.: The writhing experiment. </strong>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) <em>ko=0 m-1</em> b) <em>ko=44.8 m-1</em>.
</p>
<p align="justify">
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.
</p>
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<embed type="application/x-shockwave-flash" width="600" height="337" src="http://www.youtube.com/v/wjfUVweSEcQ"></embed>
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</p>
<p align="justify">
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, <em>1/L_gb=[EI/(ρg S)]^{-1/3}</em>, where <em>EI</em> is the bending modulus of the rod, <em>ρS</em> is its linear mass density and <em>g</em> 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].
</p>
<p>
<strong>4. Ongoing work:</strong></p>
<p align="justify">
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.
</p>
<p>
<strong>5. Concluding thoughts:</strong></p>
<p align="justify">
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.
</p>
<p>
<strong>Questions for discussion:</strong></p>
<p align="justify">
I would like to finish by opening the “i-floor” for discussion by posing a few questions:
</p>
<ul>
<li>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?</li>
<li>What modern tools (experimental, numerical and theoretical) do you use to approach this class of problems?</li>
<li>Have you come across challenges in describing the geometrically-nonlinear behavior of thin rods/filaments in your own domain?</li>
<li>Are you currently working on problems involving the mechanics of thin elastic filaments? If so, could you please share your recent work?</li>
</ul>
<p> </p>
<p align="justify">
<strong>Acknowledgments:</strong> 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.
</p>
<p>
<strong>References most directly related to this post:</strong></p>
<p align="justify">
<a href="http://web.mit.edu/preis/www/mypapers/lazarus_reis_SoftMatter_ContorningRods.pdf" title="Contorting a heavy and naturally curved elastic rod">A.Lazarus, J.T. Miller, M. Metlitz and P.M. Reis "Contorting a heavy and naturally curved elastic rod", Soft Matter, <strong>9</strong> (34), 8274 (2013). </a>
</p>
<p align="justify">
<a href="http://web.mit.edu/preis/www/mypapers/reis_JMPS_ContinuationRods_2013.pdf" title="Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method">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., <strong>61</strong>(8), 1712 (2013). </a>
</p>
<p align="justify">
For more information on research on the mechanics of slender structures from our research group - <a href="http://web.mit.edu/preis/www/" title="Pedro Reis | EGS.Lab | Elasticity, Geometry and Statistics Laboratory | MIT">EGS.Lab: Elasticity, Geometry and Statistics Laboratory</a> (MIT) - please visit: <a href="http://web.mit.edu/preis/www/" title="Pedro Reis | EGS.Lab | Elasticity, Geometry and Statistics Laboratory | MIT"> http://web.mit.edu/preis/www/</a>
</p>
<p>
<strong>Active Mechanicians working on thin rods:</strong> </p>
<p align="justify">
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):
</p>
<ul>
<li><a href="http://www.lmm.jussieu.fr/~audoly/" title="Basile Audoly (CNRS/UPMC)">Basile Audoly</a> (CNRS/UPMC),</li>
<li><a href="http://www.enm.bris.ac.uk/anm/staff/arc.html" title="Alan Champneys (Bristol)">Alan Champneys</a> (Bristol),
</li>
<li><a href="http://www.mechanics.rutgers.edu/BDC.html">Bernard Coleman</a> (Rutgers),</li>
<li><a href="http://www.damtp.cam.ac.uk/user/gold/" title="Raymond Goldstein (Cambridge)">Raymond Goldstein</a> (Cambridge),</li>
<li><a href="http://www.maths.ox.ac.uk/people/profiles/alain.goriely">Alain Goriely</a>, (Oxford),</li>
<li><a href="http://www.cs.columbia.edu/~eitan/">Eitan Grinspun</a>, (Columbia)</li>
<li><a href="http://www.math.cornell.edu/m/People/Faculty/healey">Tim Healey</a> (Cornell),</li>
<li><a href="http://www.ucl.ac.uk/~ucesgvd/" title="Gert van der Heijden (UCL)">Gert van der Heijden</a> (UCL), </li>
<li><a href="http://www.math.umbc.edu/~khoffman/">Kathleen Hoffmann</a> (Maryland)</li>
<li><a href="http://lcvmwww.epfl.ch/~jhm/" title="John Maddocks (EPFL)">John Maddocks</a> (EPFL), </li>
<li><a href="http://www.seas.harvard.edu/softmat/" title="L. Mahadevan (Harvard)">L. Mahadevan (Harvard)</a>,</li>
<li><a href="http://www.haverford.edu/math/rmanning.html">Rob Manning</a>, (Haverford),</li>
<li><a href="http://www.lmm.jussieu.fr/~neukirch/" title="Sébastien Neukirch (CNRS/UPMC)">Sébastien Neukirch</a> (CNRS/UPMC),</li>
<li><a href="http://www.me.berkeley.edu/faculty/oreilly/" title="Oliver M. O'Reilly (UC Berkeley)">Oliver O'Reilly</a> (UC Berkeley)</li>
<li> <a href="http://people.mcgill.ca/michael.paidoussis/" title="Michael P Paidoussis (McGill)">Michael P Paidoussis</a> (McGill), </li>
<li><a href="http://www.seas.upenn.edu/directory/profile.php?ID=75" title="Prashant Purohit (UPenn)">Prashant Purohit</a> (UPenn),</li>
<li><a href="http://www.math.pitt.edu/~swigon/">David Swigon</a>, (Pittsburgh)</li>
<li> <a href="http://appliedmath.arizona.edu/profile-michael-tabor" title="Michael Tabor (Arizona)">Michael Tabor</a> (Arizona), </li>
<li><a href="http://www.ucl.ac.uk/~ucess21/" title="J. Michael Thompson (UCL)">J. Michael Thompson</a> (UCL).</li>
</ul>
<p>
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.
</p>
<p>
<strong>References:</strong>
</p>
<p align="left">
[1] S. Neukirch, Phys. Rev. Lett., <strong>93</strong>, 198107 (2004).
</p>
<p align="left">
[2] N. Geblinger, A. Ismach and E. Joselevich, Nature Nanotech., <strong>3</strong>, 195 (2008).
</p>
<p align="left">
[3] K. Son, J.S. Guasto and R, Stocker, Nature Physics <strong>9</strong>, 494 (2013).
</p>
<p align="left">
[4] Y. Sun, W.M. Choi, H. Jiang, Y.Y. Huang, and J.A. Rogers, Nature Nanotech. <strong>1</strong>(3) 201 (2006).
</p>
<p align="left">
[5] A. Goriely, M. Tabor. Phys. Rev. Lett. <strong>80</strong>(7) 1564 (1998).
</p>
<p align="left">
[6] B. Audoly and Y. Pomeau. “Elasticity and geometry: from hair curls to the non-linear response of shells.” Oxford: Oxford University Press (2010).
</p>
<p align="left">
[7] S. Goyal, N. Perkins and C. Lee, Int. J. Nonlinear Mech., <strong>43</strong>, 65 (2008).
</p>
<p align="left">
[8] N. Wicks, B.L. Wardle and D. Pafitis. Int. J. of Mech. Sci. <strong>50</strong>(3) 538 (2008).
</p>
<p align="left">
[9] R. Levien, "The Elastica: A Mathematical History". UCB/EECS-2008-103, EECS Department, University of California, Berkeley.
</p>
<p align="left">
[10] E.H. Dill, Arch. Hist. Exact Sci. <strong>44</strong>(1) 1 (1992).
</p>
<p align="left">
[11] M. Davies and F. Moon, Chaos, <strong>3</strong>, 93 (1993).
</p>
<p align="left">
[12] A. Goriely and M. Tabor, Physica D 105, <strong>20</strong> (1997); A. Champneys, G. Van der Heijden, and J. Thompson, Philos. Trans. R. Soc. London, Ser. A <strong>355</strong>, 2151 (1997); P. Furrer, R. Manning, and J. Maddocks, Biophys. J. <strong>79</strong>, 116 (2000).
</p>
<p align="left">
[13] S. Karkar, R. Arquier, and B. Cochelin, User Guide MANLAB 2.0 (2011).
</p>
<p align="left">
[14] A. Lazarus, J. Miller, and P.M. Reis, J. Mech. Phys. Solids <strong>61</strong>, 1712 (2013).
</p>
<p align="left">
[15] A. Lazarus, J. Miller, M. Metlitz, and P.M. Reis, Soft Matter <strong>9</strong> (34), 8274 (2013).
</p>
<p align="left">
[16] E.J. Doedel, "AUTO: A program for the automatic bifurcation analysis of autonomous systems." Congr. Numer. <strong>30</strong> 265-284 (1981).
</p>
<p align="left">
[17] A. Greenhill, Proc. – Inst. Mech. Eng., <strong>34</strong>, 182 (1883). A. Love, "A treatise on the mathematical theory of elasticity", University Press, (1920). J. Coyne, IEEE J. Oceanic Eng.,<strong>15</strong>, 72 (1990). J. Thompson and A. Champneys, Proc. R. Soc. London, Ser. A, <strong>452</strong>, 117 (1996). A. Goriely and M. Tabor, Proc. R. Soc. London, Ser. A, <strong>454</strong>, 3183 (1998). A. Goriely and M. Tabor, Proc. R. Soc. London, Ser. A, <strong>453</strong>, 2583 (1997).
</p>
<p align="left">
[18] M. Bergou, M. Wardetzky, S. Robinson, B. Audoly and E. Grinspun "Discrete elastic rods." ACM Transactions on Graphics (TOG). <strong>27</strong>(3) 63 (2008).
</p>
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</div></div></div>Mon, 07 Oct 2013 00:00:58 +0000Pedro Reis15442 at https://imechanica.orghttps://imechanica.org/node/15442#commentshttps://imechanica.org/crss/node/15442