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25
On the Dynamics of Elastic Strips
 J. Nonlinear Sci
, 1999
"... The dynamics of elastic strips, i.e. long thin rods with noncircular crosssection, is analyzed by studying the solutions of the appropriate Kirchho# equations. First, it is shown that if a naturally straight strip is deformed into a helix, the only equilibrium helical configurations are those with ..."
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The dynamics of elastic strips, i.e. long thin rods with noncircular crosssection, is analyzed by studying the solutions of the appropriate Kirchho# equations. First, it is shown that if a naturally straight strip is deformed into a helix, the only equilibrium helical configurations are those with no internal twist and whose principal bending direction is either along the normal or the binormal. Second, the linear stability of a straight twisted strip under tension is analyzed showing the possibility of both pitchfork and Hopf bifurcations depending on the external and geometric constraints. Third, nonlinear amplitude equations are derived describing the dynamics close to the di#erent bifurcation regimes. Finally, special analytical solutions to these equations are used to describe the buckling of strips. In particular, finitelength solutions with a variety of boundary conditions are considered. KEYWORDS: Elastic Strips, Amplitude Equations, Localized solutions RUNNING TITLE: Dynami...
Nonlinear Euler buckling
, 2008
"... The buckling of hyperelastic incompressible cylindrical shells of arbitrary length and thickness under axial load is considered within the framework of nonlinear elasticity. Analytical and numerical methods for bifurcation are obtained using the Stroh formalism and the exact solution of Wilkes [Q. J ..."
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Cited by 15 (6 self)
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The buckling of hyperelastic incompressible cylindrical shells of arbitrary length and thickness under axial load is considered within the framework of nonlinear elasticity. Analytical and numerical methods for bifurcation are obtained using the Stroh formalism and the exact solution of Wilkes [Q. J. Mech. Appl. Math. 8:88–100, 1955] for the linearized problem. The main focus of this paper is the range of validity of the Euler buckling formula and its first nonlinear corrections that are obtained for thirdorder elasticity. 1
Nucleosome repositioning via loop formation
 Biophys. J
, 2003
"... ABSTRACT Active (catalyzed) and passive (intrinsic) nucleosome repositioning is known to be a crucial event during the transcriptional activation of certain eukaryotic genes. Here we consider theoretically the intrinsic mechanism and study in detail the energetics and dynamics of DNAloopmediated n ..."
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Cited by 7 (1 self)
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ABSTRACT Active (catalyzed) and passive (intrinsic) nucleosome repositioning is known to be a crucial event during the transcriptional activation of certain eukaryotic genes. Here we consider theoretically the intrinsic mechanism and study in detail the energetics and dynamics of DNAloopmediated nucleosome repositioning, as previously proposed by earlier works. The surprising outcome of the present study is the inherent nonlocality of nucleosome motion within this model–being a direct physical consequence of the loop mechanism. On long enough DNA templates the longer jumps dominate over the previously predicted local motion, a fact that contrasts simple diffusive mechanisms considered before. The possible experimental outcome resulting from the considered mechanism is predicted, discussed, and compared to existing experimental findings.
Biomimetic Propulsion for a Swimming Surgical Microrobot
 Proceedings of the 2004 IEEE International Conference on Intelligent Robots and System
, 2003
"... Abstract A surgical microrobot that swims inside the human ureter is proposed to provide a novel and minimally invasive method of kidney stone destruction. Inspired by the swimming mechanisms of bacteria such as E. coli, the robot utilizes biomimetic synthetic flagella composed of multiwalled car ..."
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Cited by 6 (2 self)
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Abstract A surgical microrobot that swims inside the human ureter is proposed to provide a novel and minimally invasive method of kidney stone destruction. Inspired by the swimming mechanisms of bacteria such as E. coli, the robot utilizes biomimetic synthetic flagella composed of multiwalled carbon nanotubes that are driven into a rotating helical shape by a micro motor. Design aspects are discussed with the focus on locomotion. The performance of the propulsion mechanism is determined through simultaneous modeling of the viscous drag on the filaments and the stress strain behavior of the nanotubes. The effects of the synthetic flagellum geometry and frequency of rotation on efficiency and swimming speed are explored. With 1nW of power, utilizing 100 Plong filaments, swimming speeds approaching 1 mm/s are shown to be possible for a realistic design. The proposed new robot would revolutionize kidney stone destruction if implemented, yet the design of the robot and the propulsion analysis are applicable to many other possible surgical procedures. I.
The mechanics of short rodlike molecules in tension
 Int. J. Nonlinear Mech
, 2008
"... The rapid development of single molecule experimental techniques in the last two decades has made it possible to accurately measure the forceextension response as well as the transverse fluctuations of individual rodlike macromolecules. This information is used in conjunction with a statistical me ..."
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Cited by 5 (1 self)
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The rapid development of single molecule experimental techniques in the last two decades has made it possible to accurately measure the forceextension response as well as the transverse fluctuations of individual rodlike macromolecules. This information is used in conjunction with a statistical mechanical model based on the treatment of the molecule as a fluctuating elastic rod to extract its bending and extension moduli. The models most commonly used to interpret the experimental data assume that the magnitude of the Brownian fluctuations are independent of the length of the macromolecule, an assumption that holds only in the asymptotic limit of infinitely long rods, and is violated in most experiments. As an alternative, we present a theoretical treatment of a finite length, fluctuating rod and determine its mechanical behavior by measuring the transverse Brownian fluctuations under the action of large stretching forces. to validate of our thoery, we have applied our methods to an experiment on short actin filaments whose forceextension relation is difficult to measure, but whose transverse deflections can be captured by current microscopy techniques. An important consequence of the short contour lengths is that the boundary conditions applied in the experiment affect the fluctuations and can no longer be neglected as is commonly done when interpreting data from forceextension measurements. Our theoretical methods account for boundary conditons and can therefore be deployed in conjuction with forceextension measurements to obtain detailed information about the mechanical response of rodlike macromolecules. 1
Plectoneme formation in twisted fluctuating rods
 J. Mech. Phys. Solids
, 2008
"... The mechanics of DNA supercoiling is a subject of crucial importance to uncover the mechanism and kinetics of several enzymes. It is therefore being investigated using several biochemical and biophysical methods including single molecule experimental techniques. An interesting problem within this r ..."
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The mechanics of DNA supercoiling is a subject of crucial importance to uncover the mechanism and kinetics of several enzymes. It is therefore being investigated using several biochemical and biophysical methods including single molecule experimental techniques. An interesting problem within this realm is that of torsional buckling and plectoneme formation in DNA as it is simultaneously put under tensile and torsional stress. Analytical solutions to this problem are difficult to find since it involves nonlinear kinematics and thermal fluctuations. In this paper we use ideas from the Kirchhoff theory of filaments to find semianalytical solutions for the average shape of the fluctuating DNA under the assumption that there is no selfcontact. The basic step in our method consists of combining a helical solution of the rod with a nonplanar localizing solution in such a way that the force, moment, position and slope remain continuous everywhere along the rod. Our solutions allow us to predict the extension vs. linking number behavior of long pieces of DNA for various values of the tension and temperature. An interesting outcome of our calculations is the prediction of a sudden change in extension at buckling which does not seem to have been emphasized in earlier theoretical models or experiments. Our predictions are amenable to falsification by recently developed single molecule techniques which can simultaneously track the forceextension as well as the torquerotation behavior of DNA.
Kovalevskaya rods and Kovalevskaya waves
, 2000
"... The Kirchhoff analogy for elastic rods establishes the equivalence between the solutions of the classical spinning top and the stationary solutions of the Kirchhoff model for thin elastic rods with circular crosssections. In this paper the Kirchhoff analogy is further generalized to show that the c ..."
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Cited by 3 (1 self)
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The Kirchhoff analogy for elastic rods establishes the equivalence between the solutions of the classical spinning top and the stationary solutions of the Kirchhoff model for thin elastic rods with circular crosssections. In this paper the Kirchhoff analogy is further generalized to show that the classical Kovalevskaya solution for the rigid body problem is formally equivalent to the solution of the Kirchhoff model for thin elastic rod with anisotropic crosssections (elastic strips). These Kovalevskaya rods are completely integrable and are part of a family of integrable traveling waves solutions for the rod (Kovalevskaya waves). The analysis of homoclinic twistless Kovalevskaya rod reveals the existence of a three parameter family of solutions corresponding to the Steklov and Bobylev integrable case of the rigid body problem. Furthermore, the existence of these integrable solutions is discussed in conjunction with recent results on the stability of strips.
Symmetry reduced dynamics of charged molecular strands
, 2010
"... The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are nonlocal when the screened Coulomb interactions, ..."
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The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are nonlocal when the screened Coulomb interactions, or LennardJones potentials between pairs of charges, are included. The nonlocal dynamics is derived in the convective representation of continuum motion by using modified EulerPoincare ́ and HamiltonPontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods in the spatial representation. The motion equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry group of the system. This approach uses the process of affine EulerPoincare ́ reduction initially developed for complex fluids. On the Hamiltonian
Compact waves on planar elastic rods
 International Journal of NonLinear Mechanics
, 2009
"... Planar Kirchhoff elastic rods with nonlinear constitutive relations are shown to admit traveling wave solutions with compact support. The existence of planar compact waves is a general property of all nonlinearly elastic intrinsically straight rods, while intrinsically curved rods do not exhibit th ..."
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Planar Kirchhoff elastic rods with nonlinear constitutive relations are shown to admit traveling wave solutions with compact support. The existence of planar compact waves is a general property of all nonlinearly elastic intrinsically straight rods, while intrinsically curved rods do not exhibit this type of behavior. It is with great pleasure that we dedicate this paper to Philippe Boulanger. Among the contributors to this volume, AG has the unique distinction to have been a student of Philippe Boulanger in both his undergraduate and graduate years, a teaching assistant for his Cours de Mécanique, a colleague in the Département de Mathématiques, a member of his Laboratoire de Mécanique, and, many years later, a collaborator.
Dynamics of charged molecular strands
, 2009
"... EulerPoincaré equations are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that the equations of motion for the dynamics of such molecular strands are ..."
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EulerPoincaré equations are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that the equations of motion for the dynamics of such molecular strands are nonlocal when the screened Coulomb interactions, or LennardJones potentials between pairs of charges are included. These nonlocal dynamical equations are derived in the convective representation of continuum motion by using modified EulerPoincaré and HamiltonPontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods in the spatial representation. The motion equations in the convective representation are shown to be affine EulerPoincaré equations relative to a certain cocycle. This property relates the geometry of the molecular strands to that of complex fluids. An