Kirchhoff Elastic Rod Research Articles

On p-Willmore disks with boundary energies

We consider an energy functional on surface immersions which includes contributions from both boundary and interior. Inspired by physical examples, the boundary is modeled as the center line of a generalized Kirchhoff elastic rod, while the interior term is arbitrarily dependent on the mean curvature and linearly dependent on the Gaussian curvature. We study equilibrium configurations for this energy in general among topological disks, as well as specifically for the class of examples known as p-Willmore energies.

First integrals for elastic curves: twisting instabilities of helices

We put forward a variational framework suitable for the study of curves whose energies depend on their bend and twist degrees of freedom. By employing the material curvatures to describe such elastic deformation modes, we derive the equilibrium equations representing the balance of forces and torques on the curve. The conservation laws of the force and torque on the curve, stemming from the Euclidean invariance of the energy, allow us to obtain first integrals of the equilibrium equations. To illustrate this framework, we apply it to determine the first integrals for isotropic and anisotropic Kirchhoff elastic rods, whose energies are quadratic in the material curvatures. We use them to analyze perturbatively the deformations of helices resulting from their twisting. We examine three kinds of twisting instabilities on unstretchable helices, characterized by their wavenumbers, depending on whether their boundaries are fixed, displaced along the radial direction or orthogonally to it. We also analyze perturbatively the effect of the bending anisotropy on the deformed states, which introduces a coupling between deformation modes with different wavenumbers.

When Is a Helix Stable?

We determine which helical equilibria of an isotropic Kirchhoff elastic rod with clamped ends are stable and which are unstable. Although the set of all helical equilibria is parametrized by four variables, with an additional fifth parameter determined by the rod's material, we show that only three of these five parameters are needed to distinguish between stable and unstable equilibria. We also show that the closure of the set of stable equilibria is star convex. With these results, we are able to compute and visualize the boundary between stable and unstable helices for the first time.

Inverse design of an isotropic suspended Kirchhoff rod: theoretical and numerical results on the uniqueness of the natural shape.

Solving the equations for Kirchhoff elastic rods has been widely explored for decades in mathematics, physics and computer science, with significant applications in the modelling of thin flexible structures such as DNA, hair or climbing plants. As demonstrated in previous experimental and theoretical studies, the natural curvature plays an important role in the equilibrium shape of a Kirchhoff rod, even in the simple case where the rod is isotropic and suspended under gravity. In this paper, we investigate the reverse problem: can we characterize the natural curvature of a suspended isotropic rod, given an equilibrium curve? We prove that although there exists an infinite number of natural curvatures that are compatible with the prescribed equilibrium, they are all equivalent in the sense that they correspond to a unique natural shape for the rod. This natural shape can be computed efficiently by solving in sequence three linear initial value problems, starting from any framing of the input curve. We provide several numerical experiments to illustrate this uniqueness result, and finally discuss its potential impact on non-invasive parameter estimation and inverse design of thin elastic rods.

Guiding Elastic Rods With a Robot-Manipulated Magnet for Medical Applications.

Magnet-tipped, elastic rods can be steered by an external magnetic field to perform surgical tasks. Such rods could be useful for a range of new medical applications because they do not require either pull wires or other bulky mechanisms that are problematic in small anatomical regions. However, current magnetic rod steering systems are large and expensive. Here, we describe a method to guide a rod using a robot-manipulated magnet located near a patient. We solve for rod deflections by combining permanent-magnet models with a Kirchhoff elastic rod model and use a resolved-rate approach to compute trajectories. Experiments show that three-dimensional trajectories can be executed accurately without feedback and that the system's redundancy can be exploited to avoid obstacles.

Robust and real-time guidewire simulation based on Kirchhoff elastic rod for vascular intervention training

Virtual reality (VR) based vascular intervention training is a fascinating innovation, which helps trainees develop skills in safety remote from patients. The vascular intervention training involves the use of flexible tipped guidewires to advance diagnostic or therapeutic catheters into a patient’s vascular anatomy. In this paper, a real-time physically-based modeling approach is proposed to simulate complicated behaviors of guidewires and catheters based on Kirchhoff elastic rod. The slender body of guidewire and catheter is simulated using more efficient special case of naturally straight, isotropic Kirchhoff rods, and the short flexible tip composed of straight or angled design is modeled using more complex generalized Kirchhoff rods. We derive the equations of motion for guidewire and catheter with continuous elastic energy, and then they were discretized using a linear implicit scheme that guarantees stability and robustness. In addition, we apply a fast-projection method to enforce the inextensibility of guidewire and catheter, while an adaptive sampling algorithm is implemented to improve the simulation efficiency without reducing accuracy. Experimental results reveal that our guidewire simulation method is both robust and efficient in a real-time performance.

A robust and real-time vascular intervention simulation based on Kirchhoff elastic rod

A virtual reality (VR) based vascular intervention simulation system is introduced in this paper, which helps trainees develop surgical skills and experience complications in safety remote from patients. The system simulates interventional radiology procedures, in which flexible tipped guidewires are employed to advance diagnostic or therapeutic catheters into vascular anatomy of a patient. A real-time physically-based modeling approach ground on Kirchhoff elastic rod is proposed to simulate complicated behaviors of guidewires and catheters. The slender body of guidewire and catheter is modeled using more efficient special case of naturally straight, isotropic Kirchhoff rods, and the shorter flexible tip composed of straight or angled design is modeled using more complex generalized Kirchhoff rods. The motion equations for guidewire and catheter were derived with continuous elastic energy, followed by a discretization using a linear implicit scheme that guarantees stability and robustness. In addition, we used a fast-projection method to enforce the inextensibility of guidewire and catheter. An adaptive sampling algorithm was also implemented to improve the simulation efficiency without decrease of accuracy. Experimental results revealed that our system is both robust and efficient in a real-time performance.

Configuration Modeling and Experimental Verification with 3D Laser Scanning Technology for a Constrained Elastica Cable

For large complex electromechanical products, cable digital design has a problem of low accuracy because of its large deformable characteristics. To solve the problem, an efficacious method is proposed to do cable geometry physical modeling, simulation and test for two-end constrained spatial flexible cables. Euler angles are used as generalized parametric variables to describe cable configuration on the base of Kirchhoff elastic rod theory. Via establishment of a cylindrical coordinate system according to Saint-Venant principle using end constraint, Euler angles’ Analytical solutions of the Kirchhoff equations are gained. Based on an open source geometry kernel system and using numerical integration ideas, a cable configuration simulation platform is developed in C++ environments. Then, a cable geometry measurement bench is founded with the use of 3D laser scan technology. Finally, a cable configuration simulation and experimental example is given to verify the accuracy of the proposed model and relative algorithm, through 3D positional deviation analysis.

Extendability of Kirchhoff elastic rods in complete Riemannian manifolds

The Kirchhoff elastic rod is a classical mathematical model of equilibrium configurations of thin elastic rods, and is defined to be a solution of the Euler-Lagrange equations associated to the energy with the effect of bending and twisting. We consider the initial-value problem for the Euler-Lagrange equations in a Riemannian manifold. In a previous paper, the author proved the existence and uniqueness of global solutions of the initial-value problem in the case where the ambient space is a space form. In the present paper, we extend this result to the case where the ambient space is a general complete Riemannian manifold. This implies that an arbitrary Kirchhoff elastic rod of finite length in a complete Riemannian manifold extends to that of infinite length.

Curvature Elastic Energy Model for Carbon Nanotubes

Since carbon nanotubes (CNTs) were discovered, due to their unique and novel physical and chemical characteristics, many studies focus on them. In this article, a two-dimensional (2D) curvature elastic energy model for isotropic tube is presented, and reduced to a one-dimensional (1D) continuous model which is in accordance with Kirchhoff elastic rod theory. The problems remaining to be solved are discussed.

Singular inextensible limit in the vibrations of post-buckled rods: Analytical derivation and role of boundary conditions

In-plane vibrations of an elastic rod clamped at both extremities are studied. The rod is modeled as an extensible planar Kirchhoff elastic rod under large displacements and rotations. Equilibrium configurations and vibrations around these configurations are computed analytically in the incipient post-buckling regime. Of particular interest is the variation of the first mode frequency as the load is increased through the buckling threshold. The loading type is found to have a crucial importance as the first mode frequency is shown to behave singularly in the zero thickness limit in the case of prescribed axial displacement, whereas a regular behavior is found in the case of prescribed axial load.

Kirchhoff elastic rods in five-dimensional space forms whose centerlines are not helices

We give examples of Kirchhoff rod centerlines in five-dimensional space forms which are fully immersed and not helices. We also show that the natural curvatures of these Kirchhoff rod centerlines are expressed explicitly in terms of Jacobi sn function.

Effect of the boundary conditions and influence of the rotational inertia on the vibrational modes of an elastic ring.

We present the small-amplitude vibrations of a circular elastic ring with periodic and clamped boundary conditions. We model the rod as an inextensible, isotropic, naturally straight Kirchhoff elastic rod and obtain the vibrational modes of the ring analytically for periodic boundary conditions and numerically for clamped boundary conditions. Of particular interest are the dependence of the vibrational modes on the torsional stress in the ring and the influence of the rotational inertia of the rod on the mode frequencies and amplitudes. In rescaling the Kirchhoff equations, we introduce a parameter inversely proportional to the aspect ratio of the rod. This parameter makes it possible to capture the influence of the rotational inertia of the rod. We find that the rotational inertia has a minor influence on the vibrational modes with the exception of a specific category of modes corresponding to high-frequency twisting deformations in the ring. Moreover, some of the vibrational modes over or undertwist the elastic rod depending on the imposed torsional stress in the ring.

Quasi-static manipulation of a Kirchhoff elastic rod based on a geometric analysis of equilibrium configurations

Consider a thin, flexible wire of fixed length that is held at each end by a robotic gripper. Any curve traced by this wire when in static equilibrium is a local solution to a geometric optimal control problem, with boundary conditions that vary with the position and orientation of each gripper. We prove that the set of all local solutions to this problem over all possible boundary conditions is a smooth manifold of finite dimension that can be parameterized by a single chart. We show that this chart makes it easy to implement a sampling-based algorithm for quasi-static manipulation planning. We characterize the performance of such an algorithm with experiments in simulation.

Static and dynamic stability results for a class of three-dimensional configurations of Kirchhoff elastic rods

We analyze the dynamical stability of a naturally straight, inextensible and unshearable elastic rod, under tension and controlled end rotation, within the Kirchhoff model in three dimensions. The cases of clamped boundary conditions and isoperimetric constraints are treated separately. We obtain explicit criteria for the static stability of arbitrary extrema of a general quadratic strain energy. We exploit the equivalence between the total energy and a suitably defined norm to prove that local minimizers of the strain energy, under explicit hypotheses, are stable in the dynamic sense due to Liapounov. We also extend our analysis to damped systems to show that static equilibria are dynamically stable in the Liapounov sense, in the presence of a suitably defined local drag force.

Analytical reduction of the non-circular Kirchhoff elastic rod model with the periodically varying bending rigidities

The analytical equation of the non-circular elastic rod governed by the low-dimensional and single-variable system was obtained. We reconstructed the Kirchhoff equations with a group of complex vectors and then introduced a novel complex variable solution of the torque. The knowledge of effective bending rigidity was further extended to fit the characteristic of non-circular cross section with periodically varying bending coefficients embodied in the sequence-dependent effects of a DNA molecule. The resulting low-dimensional systems were closely fit for the analytical analysis, so that the numerical simulation turned out to be not an exclusive approach during the analysis of the non-circular cross section Kirchhoff model, such as deriving the asymptotic solutions and microscopic topology of the spatial configuration.

Morphoelastic rods. Part I: A single growing elastic rod

A theory for the dynamics and statics of growing elastic rods is presented. First, a single growing rod is considered and the formalism of three-dimensional multiplicative decomposition of morphoelasticity is used to describe the bulk growth of Kirchhoff elastic rods. Possible constitutive laws for growth are discussed and analysed. Second, a rod constrained or glued to a rigid substrate is considered, with the mismatch between the attachment site and the growing rod inducing stress. This stress can eventually lead to instability, bifurcation, and buckling.

GLOBAL SOLUTIONS OF THE EQUATION OF THE KIRCHHOFF ELASTIC ROD IN SPACE FORMS

AbstractThe Kirchhoff elastic rod is one of the mathematical models of equilibrium configurations of thin elastic rods, and is defined to be a solution of the Euler–Lagrange equations associated to the energy with the effect of bending and twisting. In this paper, we consider Kirchhoff elastic rods in a space form. In particular, we give the existence and uniqueness of global solutions of the initial-value problem for the Euler–Lagrange equations. This implies that an arbitrary Kirchhoff elastic rod of finite length extends to that of infinite length.

A one-dimensional continuous model for carbon nanotubes

The continuous two-dimensional (2D) elastic model for single-walled carbon nanotubes (SWNTs) provided by Tu and Ou-Yang in [Phys. Rev. B \textbf{65}, 235411 (2003)] is reduced to a one-dimensional (1D) curvature elastic model strictly. This model is in accordance with the isotropic Kirchhoff elastic rod theory. Neglecting the in-plane strain energy in this model, it is suitable to investigate the nature features of carbon nanotubes (CNTs) with large deformations and can reduce to the string model in [Phys. Rev. Lett. \textbf{76}, 4055 (1997)] when the deformation is small enough. For straight chiral shapes, this general model indicates that the difference of the chiral angle between two equilibrium states is about $\pi/6$, which is consistent with the lattice model. It also reveals that the helical shape has lower energy for per atom than the straight shape has in the same condition. By solving the corresponding equilibrium shape equations, the helical tube solution is in good agreement with the experimental result, and super helical shapes are obtained and we hope they can be found in future experiments.

Vibrations of post-buckled rods: The singular inextensible limit

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 the variation of mode frequencies as the load is increased through the buckling threshold. While for some modes there are no qualitative changes in the mode frequencies, other frequencies experience rapid variations after the buckling threshold, the thinner the rod, the more abrupt the variations. Eventually, a mismatch for half of the frequencies at buckling arises between the zero thickness limit of the extensible model and the inextensible model.