Sebastien Neukirch

• Sorbonne University

Thin elastic ribbons represent a class of intermediary objects lying in-between thin elastic plates and thin elastic rods. Although the two latter families of thin structures have received much interest from the Computer Graphics community over the last decades, ribbons have seldom been considered and modelled numerically so far, in spite of a grow...

Mechanical stress and conformation of helical elastic rods clamped at both ends were studied upon unwinding. By axial rotation of one end, the winding number was progressively changed from the natural one (n=n0) to complete chirality inversion (n=−n0) while keeping the total elongation fixed and monitoring the applied torque M and tension T. Along...

Starting from the theory of elastic plates, we derive a non-linear one-dimensional model for elastic ribbons with thickness t, width a and length ℓ, assuming t≪a≪ℓ. It takes the form of a rod model with a specific non-linear constitutive law accounting for both the stretching and the bending of the ribbon mid-surface. The model is asymptotically co...

We introduce a selected set of protocols inspired from the Soft Matter Physics community in order to validate Computer Graphics simulators of slender elastic structures possibly subject to dry frictional contact. Although these simulators were primarily intended for feature film animation and visual effects, they are more and more used as virtual d...

International audience We compare different models describing the buckling, post-buckling and vibrations of elastic beams in the plane. Focus is put on the first buckled equilibrium solution and the first two vibration modes around it. In the incipient post-buckling regime, the classic Woinowsky-Krieger model is known to grasp the behavior of the s...

We propose a robust and efficient numerical model to compute stable equilibrium configurations of clamped elastic ribbons featuring arbitrarily curved natural shapes. Our spatial discretization scheme relies on elements characterized by a linear normal curvature and a quadratic geodesic torsion with respect to arc length. Such a high-order discreti...

Silicone elastomers such as polydimethylsiloxane (PDMS) are convenient materials routinely used in laboratories that combine ease of preparation, flexibility, transparency and gas permeability. However, these elastomers are known to contain a small fraction of uncrosslinked low-molecular-weight oligomers, the effects of which are not completely und...

A challenge in soft robotics and soft actuation is the determination of an elastic system which spontaneously recovers its trivial path during postcritical deformation after a bifurcation. The interest in this behaviour is that a displacement component spontaneously cycles around a null value, thus producing a cyclic soft mechanism. An example of s...

Reserving the right to stretch Retractable antennae or certain spider silks can stretch well beyond their apparent length because they have a reserve of material that lets them expand and contract over much longer distances. Grandgeorge et al. made nonwoven fibrous membranes by electrospinning a block copolymer with varying ratios of two components...

We study an elastic rod bent into an open trefoil knot and clamped at both ends. The question we consider is whether there are stable configurations for which there are no points of self-contact. This idea can be fairly easily replicated with a thin strip of paper, but is more difficult or even impossible with a flexible wire. We search for such co...

A liquid drop sitting on an elastic rod may act as a winch, or windlass, and pull the rod inside itself and coil it. This windlass effect has been shown to be generated by surface tension forces and to work best for small systems. Here we study the case where the drop is large enough so that its weight interferes with surface tension and modifies t...

Soft stretchable materials are key for arising technologies such as stretchable electronics or batteries, smart textiles, biomedical devices, tissue engineering and soft robotics. Recent attempts to design such materials, via e.g. micro-patterning of wavy fibres on soft substrates, polymer engineering at the molecular level or even kirigami techniq...

A flexible fiber carrying a liquid drop may coil inside the drop thereby creating a drop-on-fiber system with an ultra-extensible behavior. During compression, the excess fiber is spooled inside the droplet and capillary forces keep the system taut. During subsequent elongation, the fiber is gradually released and if a large number of spools is unc...

We demonstrate the impressive adhesive qualities of Uloborid spider orbweb capture when dry, which are lost when the nano-filament threads are wetted. A force sensor with a 50 nN–1mN detection sensitively allowed us to measure quantitatively the stress–strain characteristics of native silk threads in both the original dry state and after wetting by...

We report an unexpected behavior in wetting dynamics on soft silicone substrates: the dynamics of aqueous droplets deposited on vertical plates of such elastomers exhibits two successive speed regimes. This macroscopic observation is found to be closely related to microscopic phenomena occurring at the scale of the polymer network: we show that unc...

Capillary forces acting at the surface of a liquid drop can be strong enough to deform small objects and recent studies have provided several examples of elastic instabilities induced by surface tension. We present such an example where a liquid drop sits on a straight fiber, and we show that the liquid attracts the fiber which thereby coils inside...

We investigate the mechanics of elastic fibres carrying liquid droplets. In such systems, buckling may localize inside the drop cavity if the fibre is thin enough. This so-called drop-on-coilable-fibre system exhibits a surprising liquid-like response under compression, and a solid-like response under tension. Here we analyze this unconventional be...

DOI:https://doi.org/10.1103/PhysRevE.95.019905

A challenge in soft robotics and soft actuation is the determination of an elastic system that spontaneously recovers its trivial path during postcritical deformation after a bifurcation. The interest in this behavior is that a displacement component spontaneously cycles around a null value, thus producing a cyclic soft mechanism. An example of suc...

We observe and investigate an unexpected behavior in the dynamics of aqueous droplets sliding down on vertical plates of soft silicone elastomers, where two successive velocity regimes are present. This macroscopic observation is found to be closely related to microscopic phenomena at the scale of the polymer network: we demonstrate that uncrosslin...

Significance The spiraling capture threads of spider orb webs are covered with thousands of tiny glue droplets whose primary function is to entrap insects. In this paper we demonstrate that the function of the drops goes beyond that of gluing prey for they also play a role in the mechanical properties of these fibers—usually ascribed solely to the...

A measure of the writhing of a curve is introduced and is used to extend the CǍlugǍreanu decomposition for closed curves, as well as the polar decomposition for curves bound between planes. The new writhe measure is also shown to be able to assess changes in linking due to belt-trick and knotting type deformations, and further its utility is illust...

We demonstrate the impressive adhesive qualities of uloborid spider orb-web capture when dry, which are lost when the nano-filament threads are wetted. A force sensor with a 50 nN-1 mN detection sensitively allowed us to measure quantitatively the stress-strain characteristics of native silk threads in both the original dry state and after wetting...

Motivated by recent experimental observations of capillary-induced spooling of fibers inside droplets both in spider capture silk and in synthetic systems, we investigate the behavior of a fiber packed in a drop. Using a simplified 2D model, we provide analytical predictions for the buckling threshold and the deep post-buckling asymptotic behavior....

Spiders' webs and gossamer threads are often paraded as paradigms for lightweight structures and outstanding polymers. Probably the most intriguing of all spider silks is the araneid capture thread, covered with tiny glycoprotein glue droplets. Even if compressed, this thread remains surprisingly taut, a property shared with pure liquid films, allo...

These notes give a short introduction to the methods for the study of stability of elastic structures. We consider only the finite-dimensional case, where the state of the system is represented by a discrete set of variables. The core of the exposition focuses on the illustration of energetic methods where equilibrium and stability are found by stu...

The beam on elastic foundation is a general model used in physical, biological, and technological problems to study delamination, wrinkling, or pattern formation. Recent focus has been given to the buckling of beams deposited on liquid baths, and in the regime where the beam is soft compared to hydrostatic forces the wrinkling pattern observed at b...

We report on the capillarity-induced snapping of elastic beams. We show that a millimeter-sized water drop gently deposited on a thin buckled polymer strip may trigger an elastocapillary snap-through instability. We investigate experimentally and theoretically the statics and dynamics of this phenomenon and we further demonstrate that snapping can...

We study the interaction of an elastic beam with a liquid drop in the case where bending and extensional effects are both present. We use a variational approach to derive equilibrium equations and constitutive relation for the beam. This relation is shown to include a term due to surface energy in addition to the classical Young's modulus term, lea...

We present a free energy model for structural transitions of the DNA double helix driven by tensile and torsional stress. Our model is coarse grained and is based on semiflexible polymer descriptions of B-DNA, underwound L-DNA, and highly overwound P-DNA. The statistical-mechanical model of plectonemic supercoiling previously developed for B-DNA is...

The snap-through instability, which is present in a wide range of systems ranging from carnivorous plants to MEMS, is a well-known phenomenon in solid mechanics : when a buckled elastic beam is subjected to a transverse force, above a critical load value the buckling mode is switched. Here, we revisit this phenomenon by studying snap-through under...

The small-amplitude in-plane vibrations of an elastic rod clamped at both extremities are studied. The rod is modeled as an extensible, shearable, planar Kirchhoff elastic rod under large displacements and rotations, and the vibration frequencies are computed both analytically and numerically as a function of the loading. Of particular interest is...

Recent single-molecule experiments have observed that formation of a plectonemically supercoiled region in a stretched, twisted DNA proceeds via abrupt formation of a small plectonemic "bubble." A detailed mesoscopic model is presented for the formation of plectonemic domains, including their positional entropy, and the influence of small chiral lo...

How many points in space are needed to define a circular helix? We show here that given 3 distinct points in space there exist continuous families of helices passing through these points. Given 4 generic distinct points there is no helix. However, a discrete family of helices through 3 points can be specified if an additional property of the helix...

A drop impacting a target cutout in a thin polymer film is wrapped by the film in a dynamic sequence involving both capillary forces and inertia. Different 3D structures can be produced from a given target by slightly varying the impact parameters. A simplified model for a nonlinear dynamic Elastica coupled with a drop successfully explains this sh...

We study the mixture of extended and supercoiled DNA that occurs in a twisted DNA molecule under tension. Closed-form asymptotic solutions for the supercoiling radius, extension, and torque of the molecule are obtained in the high-force limit where electrostatic and elastic effects dominate. We demonstrate that experimental data obey the extension...

When a flexible material is placed in contact with a liquid-air interface, capillary forces may cause deformations and large displacements in the structure. Such kind of elastocapillary interactions play a crucial role in many technological applications, like deflection of nanotubes carpets or microscale self-assembly. We study the problem of a dro...

An elastica buckled in the form of an arch is subjected to a transverse force. Above a critical load value, the buckling mode is switched and the elastica takes the form of a reversed arch. This is the well-known snap-through phenomenon which has been extensively studied in solid mechanics. Here, we revisit this phenomenon and show that capillary f...

A liquid drop impacting a thin elastic membrane forms a "dynamical capillary origami" on the very rapid capillary timescale. Dynamics is here a key ingredient that allows for shape selection of the elastocapillary bundle based only on the impact velocity. We study this phenomenon using a simplified 2D setup, where a drop impacts a narrow polymer st...

We present a self-contained theory for the mechanical response of DNA in extension–rotation single molecule experiments. The theory is based on the elasticity of the double-helix and the electrostatic repulsion between two DNA duplex. The configuration of the molecule at large imposed rotation is assumed to comprise two phases, linear and superheli...

Extension jumps were recently observed in single-molecule experiments where a DNA molecule (few kbp long) is held under tension while its ends are slowly rotated. For low rotation the molecule is believed to adopt (disordered) straight configurations and when a rotation threshold is reached the molecule jumps into a supercoiled phase: plectonemes a...

We use the existing data of force-extension experiments on F-actin molecules tied into knots to compute a value of 0.15 for the static friction coefficient for contact between different parts of the same molecule with itself. This estimate for protein-protein friction is relevant for the stabilization of the 273 known proteins with knots, one perce...

Ces dernières années ont vu un intérêt grandissant des physiciens pour des problèmes de mécanique : hydrodynamique, turbulence, fracture, tribologie, milieux granulaires... En effet, ces domaines recèlent de beaux problèmes d’instabilités, d’effets non linéaires ou de singularités. Les jeunes générations de physiciens et de mécaniciens ont convergé...

We persist in considering that, for a wide range of (experimentally available) forces and torques, evaluating the writhe of a DNA molecule in magnetic tweezers experiments should not be done with Fuller's formula. We propose a tentative plot of the limit of applicability of Fuller's formula in the (force, torque) plane.

We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically non-linear. The probl...

This dissertation deals with equilibrium, stability and vibrations of twisted rods. First the model used (i.e. the Kirchhoff equations) is presented from two perspectives : (i) as a direct theory, the special Cosserat theory of rods, and (ii) as a 3D -> 1D asymptotic theory. The core of the text is dedicated to a relatively comprehensive study of e...

We present a self-contained theory for the mechanical response of DNA in single molecule experiments. Our model is based on a one-dimensional continuum description of the DNA molecule and accounts both for its elasticity and for DNA-DNA electrostatic interactions. We consider the classical loading geometry used in experiments where one end of the m...

A polyhelix is continuous space curve with continuous Frenet frame that consists of a sequence of connected helical segments. The main result of this paper is that given n points in space, there exist infinitely many polyhelices passing through these points. These curves are by construction continuous with continuous derivatives and are completely...

In the ATP synthase, transmission of energy from the membrane-embedded F0 sector to the catalytic F1 sector is accomplished by two stalks composed of coiled-coils. The great efficiency of the enzyme, despite the presence of a symmetry mismatch between the F1 and F0 sectors, suggests the involvement of elastic elements that store energy during the c...

The linking and writhing numbers are key quantities when characterizing the structure of a piece of supercoiled DNA. Defined as double integrals over the shape of the double helix, these numbers are not always straightforward to compute, though a simplified formula was established in a theorem by Fuller [Proc. Natl. Acad. Sci. U.S.A. 75, 3557 (1978...

A drop falling on a thin elastic sheet is rapidly trapped after impact by self-folding of the sheet around the drop. This trapping process, due to capillary forces, occurs on the fast timescale of hydrophobic rebound. The resulting packed drop presents a complex three-dimensional shape, characteristic of the interplay between elasticity and capilla...

Abstract: We consider an elastic rod model for twisted DNA in the plectonemic regime. The molecule is treated as an impenetrable tube with an effective, adjustable radius. The model is solved analytically, and we derive formulas for the contact pressure, twisting moment, and geometrical parameters of the supercoiled region. We apply our model to ma...

Understanding the three-dimensional structure of proteins is critical to understand their function. While great progress is being made in understanding the structures of soluble proteins, large classes of proteins such as membrane proteins, large macromolecular assemblies, and partially organized or heterogeneous structures are being comparatively...

In a recent paper, we derived a solution to the Kirchhoff equations representing a knotted elastic rod held by a tensile force applied at its ends. This problem has been formulated as the minimization of a curvature energy in the presence of a topological constraint. We extend this analysis to the case of a knot subjected to both a tensile force an...

Coiled coils are important protein-protein interaction motifs with high specificity that are used to assemble macromolecular complexes. Their simple geometric organization, consisting of alpha helices wrapped around each other, confers remarkable mechanical properties. A geometrical and mechanical continuous model taking into account sequence effec...

We consider an elastic rod model for twisted DNA in the plectonemic regime. The molecule is treated as an impenetrable tube with an effective, adjustable radius. The model is solved analytically and we derive formulas for the contact pressure, twisting moment and geometrical parameters of the supercoiled region. We apply our model to magnetic tweez...

The DNA molecule is modeled as an elastic rod with bending and twisting rigidities, subjected to external tension and twist applied at one end, the other end being clamped. We study the plectonemic equilibrium of such a rod, taking into account the impenetrability constraint. Numerical solutions of this boundary value problem have previously shown...

We study the mechanical response of elastic rods bent into open knots, focusing on the case of trefoil and cinquefoil topologies. The limit of a weak applied tensile force is studied both analytically and experimentally: the Kirchhoff equations with self-contact are solved by means of matched asymptotic expansions; predictions on both the geometric...

When a thin elastic structure comes in contact with a liquid interface, capillary forces can be large enough to induce elastic deformations. This effect becomes particularly relevant at small scales where capillary forces are predominant, for example in microsystems (micro-electro-mechanical systems or microfluidic devices) under humid environments...

Les forces capillaires exercées sur une structure élastique élancée peuvent être suffisantes pour déformer cette structure. Ces effets capillaires deviennent prépondérants à petite échelle, par exemple dans les systèmes micro- fluidiques ou les micro-systèmes mécaniques (MEMs). Nous étudions expérimentalement un système modèle macroscopique où vont...

Twining plants achieve vertical growth by revolving around supports of different sizes on which they exert a pressure. This observation raises many intriguing questions that are addressed within the framework of elastic filamentary structures by modeling the stem close to the apex as a growing elastic rod. The analysis shows that vertical growth is...

We give the results of large deflection experiments involving the bending and twisting of 1 mm diameter nickel-titanium alloy rods, up to 2 m in length. These results are compared to calculations based on the Cosserat theory of rods. We present details of this theory, formulated as a boundary value problem. The mathematical boundary conditions mode...

When thin brittle rods such as dry spaghetti pasta are bent beyond their limit curvature, they often break into more than two pieces, typically three or four. With the aim of understanding these multiple breakings, we study the dynamics of a bent rod that is suddenly released at one end. We find that the sudden relaxation of the curvature at this e...

An elastic model that specifically included DNA twist rigidity extraction from supercoiling data but left out thermal fluctuations, was presented. The supercoiling response of overtwisted DNA was reproduced quantitatively and an estimate of the effective supercoiling radius was also obtained. The effective supercoiling radius p and the twist rigidi...

We consider equilibrium configurations of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods when subject to end loads and clamped boundary conditions. In a first paper [Neukirch & Henderson, 2002], we discussed symmetry properties of the equilibrium configurations of the center line of the rod. Here, we are int...

We use an elastic rod model with contact to study the extension versus rotation diagrams of single supercoiled DNA molecules. We reproduce quantitatively the supercoiling response of overtwisted DNA and, using experimental data, we get an estimation of the effective supercoiling radius and of the twist rigidity of B-DNA. We find that unlike the ben...

We use the Cosserat rod theory to present a unified picture of jump phenomena, associated with looping, snap-through, pop-out, etc., in twisted clamped rods undergoing large deflections. Both contact-free rods and rods with isolated points of self-contact are considered. Taking proper account of the symmetries of the problem we find that an arbitra...

We show how an energy analysis can be used to derive the equilibrium equations and boundary conditions for an end-loaded variable ply much more efficiently than in previous works. Numerical results are then presented for a clamped balanced ply approaching lock-up. We also use the energy method to derive the equations for a more general ply made of...

We study the mechanics of uniform n-plies, correcting and extending previous work in the literature. An n-ply is the structure formed when n pretwisted strands coil around one another in helical fashion. Such structures are encountered widely in engineering (mooring ropes, power lines) and biology (DNA, proteins). We first show that the well-known...

We investigate the configurations of twisted elastic rods under applied end loads and clamped boundary conditions. We classify all the possible equilibrium states of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods. We show that all solutions of the clamped boundary value problem exhibit a π-flip symmetry. The...

We use three different approaches to describe the static spatial configurations of a twisted rod as well as its stability during rigid loading experiments. The first approach considers the rod as infinite in length and predicts an instability causing a jump to self-contact at a certain point of the experiment. Semi-finite corrections, taken into ac...

In this paper we address the mechanics of ply formation in DNA supercoils. We extend the variable ply formulation of Coleman & Swigon to include end loads, and the derived constitutive relations of this generalized ply are shown to be in excellent agreement with experiments. We make a careful physical examination of the uniform ply in which two str...

We study a dissipative dynamical system that models a parametric instability in a plasma. This instability is due to the interaction of a whistler with the ion acoustic wave and a plasma oscillation near the lower hybrid resonance. The amplitude of these three oscillations obey a three-dimensional system of ordinary differential equations which exh...

We introduce a method to bound attractors of dissipative dynamical systems in phase and parameter spaces. The method is based on the determination of families of transversal surfaces (surfaces crossed by the flow in only one direction). This technique yields very restrictive geometric bounds in phase space for the attractors. It also gives ranges o...

Weintroduce a new method to bound attractors of dissipative dynamical systems in phase and parameters spaces. The method is based on the determination of families of transversal surfaces #surfaces crossed by the #ow in only one direction#. This technique yields very restrictive geometric bounds in phase space for the attractors. It also gives range...

Les 7 membres du jury : Referee (see report): Freddy Dumortier, Professeur Limburgs Universitair Centrum Universitaire Campus B-3590 Diepenbeek,Belgique 32 11 26 80 04 fdumorti@luc.ac.be Referee (see report): Jaume Llibre, Professeur Departament de Matematiques Universitat autonoma de Barcelona 08193 Bellaterra, Barcelona, Espagne jllibre@mat.uab.e...

In recent papers we have introduced a method for the study of limit cycles of the Lienard system: dot{x}=y-F(x), dot{y}=-x, where F(x) is an odd polynomial. The method gives a sequence of polynomials R_n(x), whose roots are related to the number and location of the limit cycles, and a sequence of algebraic approximations to the bifurcation set of t...

In this paper, we study the bifurcation of limit cycles in Lienard systems of the form dot(x)=y-F(x), dot(y)=-x, where F(x) is an odd polynomial that contains, in general, several free parameters. By using a method introduced in a previous paper, we obtain a sequence of algebraic approximations to the bifurcation sets, in the parameter space. Each...

In this paper, we study a Li\'enard system of the form $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}=y$-${}F(x), \stackrel{\ifmmode \dot{}\else \.{}\fi{}}{y}=$-${}x$, where $F(x)$ is an odd polynomial. We introduce a method that gives a sequence of algebraic approximations to the equation of each limit cycle of the system. This sequence seems to co...

In this paper, we consider three-dimensional dynamical systems, as for example the Lorenz model. For these systems, we introduce a method for obtaining families of two-dimensional surfaces such that trajectories cross each surface of the family in the same direction. For obtaining these surfaces, we are guided by the integrals of motion that exist...

When a thin elastic structure comes in contact with a liquid interface, capillary forces can be large enough to induce elastic deformations. This effect becomes particularly relevant at small scales where capillary forces are predominant, for example in microsystems (Micro-Electro-Mechanical Systems or microfluidic devices) under hu- mid environmen...

When twining plants grow they revolve around a support on which they exert a force in order to achieve vertical growth. In 1865 Charles Darwin realized that twining plants cannot grow on supports that are too wide. Here, mechanical aspects of this problem are investigated by modeling the stem close to the apex as a growing planar elastic rod with i...

Nous calculons la réponse mécanique d'une tige élastique nouée sous la forme d'un noeud de trèfle ouvert. Dans la limite de faibles forces de tension appliquées les équations de Kirchhoff en présence de contact de la tige avec elle même sont résolues par une méthode de développement asymptotique. Les predictions sur la géométrie du noeud sont confr...