Inextensible Rod Research Articles

Infinitesimal deformations and stability of rods made of nonlocal elastic materials

Aim of the paper is the formulation of a criterion of infinitesimal stability for a class of rods made of nonlocal elastic materials. To that end, the nonlinear equilibrium equations of naturally straight, inextensible rods subject to terminal loads are written, and the constitutive equation assumed to represent the material response in rods of finite length is discussed. Then, the equations describing the infinitesimal deformations superimposed upon a finite one are deduced. The expression of the work done by the increments of the external loads associated with an infinitesimal deformation is employed to formulate, for the considered rods, the criterion of infinitesimal stability which does not require the existence of a stored-energy function. The criterion is applied to study the stability of simply supported rods subject to axial forces, of rods with one end clamped and the other one constrained to have the tangent parallel to the undeformed rod axis, subject to axial forces and twisting couples, and of annular rods formed from naturally straight rods with the addition of twist. The results show that rods made of nonlocal elastic materials exhibit a reduction in rigidity with respect to rods having the same geometry and made of usual elastic materials with the same tensile and shear moduli.

Nonlinear Oscillations Induced by Follower Forces in Prestressed Clamped Rods Subjected to Drag

Elastic-driven slender filaments subjected to compressive follower forces provide a synthetic way to mimic the oscillatory beating of biological flagella and cilia. Here, we use a continuum model to study the dynamical, nonlinear buckling instabilities that arise due to the action of nonconservative follower forces on a prestressed slender rod clamped at both ends and allowed to move in a fluid. Stable oscillatory responses are observed as a result of the interplay between the structural elastic instability of the inextensible slender rod, geometric constraints that control the onset of instability, energy pumped into the system by the active follower forces, and motion-driven fluid dissipation. Initial buckling instabilities are initiated by the effect of the follower forces and inertia; fluid drag subsequently allows for the active energy pumped into the system to be dissipated away and results in self-limiting amplitudes. By integrating the equations of equilibrium and compatibility conditions with linear constitutive laws, we compute the critical follower forces for the onset of oscillations, emergent frequencies of these solutions, and the postcritical nonlinear rod shapes for two forms of the drag force, namely linear Stokes drag and quadratic Morrison drag. For a rod with fixed inertia and drag parameters, the minimum (critical) force required to initiate stable oscillations depends on the initial slack and weakly on the nature of the drag force. Emergent frequencies and the amplitudes postonset are determined by the extent of prestress as well as the nature of the fluid drag. Far from onset, for large follower forces, the frequency of the oscillations can be predicted by evoking a power balance between the energy input by the active forces and the dissipation due to fluid drag.

Post-Buckling Analysis of a Rod Confined in a Cylindrical Tube

Understanding the buckling and post-buckling behavior of rods confined in a finite space is of both scientific and engineering significance. Under uniaxial compression, an initially straight and slender rod confined in a tube may buckle into a sinusoidal shape and subsequently evolve into a complicated, three-dimensional (3D) helical shape. In this paper, we combine theoretical and numerical methods to investigate the post-buckling behavior of confined rods. Two theoretical models, which are based on the inextensible and extensible rod theories, respectively, are proposed to derive the analytical expressions for the axial compressive stiffness in the sinusoidal post-buckling stage. The former is concise in formulation and can be easily applied in engineering, while the latter works well in a broader scope of post-buckling analysis. Both methods can give a good approximation to the sinusoidal post-buckling stiffness and the former model is proved to be a zeroth-order approximation of the latter. The flexible multibody dynamics method based on the Timoshenko's geometrically exact beam theory is used to examine the accuracy of the two models. The methods presented in this work can be used in, for example, drilling engineering in oil and gas industries.

Direct Position‐Based Solver for Stiff Rods

AbstractIn this paper, we present a novel direct solver for the efficient simulation of stiff, inextensible elastic rods within the position‐based dynamics (PBD) framework. It is based on the XPBD algorithm, which extends PBD to simulate elastic objects with physically meaningful material parameters. XPBD approximates an implicit Euler integration and solves the system of non‐linear equations using a non‐linear Gauss–Seidel solver. However, this solver requires many iterations to converge for complex models and if convergence is not reached, the material becomes too soft. In contrast, we use Newton iterations in combination with our direct solver to solve the non‐linear equations which significantly improves convergence by solving all constraints of an acyclic structure (tree), simultaneously. Our solver only requires a few Newton iterations to achieve high stiffness and inextensibility. We model inextensible rods and trees using rigid segments connected by constraints. Bending and twisting constraints are derived from the well‐established Cosserat model. The high performance of our solver is demonstrated in highly realistic simulations of rods consisting of multiple 10 000 segments. In summary, our method allows the efficient simulation of stiff rods in the PBD framework with a speedup of two orders of magnitude compared to the original XPBD approach.

Exact equilibrium solutions for nonlinear spatial deformations of nanorods with application to buckling under terminal force and couple

The paper is concerned with nonlinear spatial deformations of nanorods, described by exact solutions of the equilibrium equations of a rod theory based on the kinematics of Kirchhoff’s inextensible rods and the constitutive equation of Eringen’s nonlocal materials. The nonlinear equilibrium equations of nonlocal elastic rods that are prismatic in a natural state are established and their solutions are obtained for both configurations in which the axial curve has constant curvature and torsion, and general configurations in which the axial curve is described by means of elliptic functions. As an example of application of the theory, the buckling in space under terminal force and couple of nonlocal rods with one end clamped and the other one constrained to have the tangent parallel to the direction of the undeformed rod axis, is examined. For such rods, the eigencurves corresponding to bifurcation from configurations in which the axial curve is straight are found, and the exact solutions for the post-buckling configurations are presented. The results show that, along bifurcating branches on which the axial force is constant the rods, through a sequence of spatial equilibrium configurations, arrive at configurations in which the axial curve is a (planar) non-inflexional elastica. The comparison of the behavior of rods made of usual and nonlocal elastic materials shows that the presence of a nonlocal material has effects analogous to those due to a reduction in the rigidity of the rods.

VCSim3: a VR simulator for cardiovascular interventions

PurposeEffective and safe performance of cardiovascular interventions requires excellent catheter/guidewire manipulation skills. These skills are currently mainly gained through an apprenticeship on real patients, which may not be safe or cost-effective. Computer simulation offers an alternative for core skills training. However, replicating the physical behaviour of real instruments navigated through blood vessels is a challenging task.MethodsWe have developed VCSim3—a virtual reality simulator for cardiovascular interventions. The simulator leverages an inextensible Cosserat rod to model virtual catheters and guidewires. Their mechanical properties were optimized with respect to their real counterparts scanned in a silicone phantom using X-ray CT imaging. The instruments are manipulated via a VSP haptic device. Supporting solutions such as fluoroscopic visualization, contrast flow propagation, cardiac motion, balloon inflation, and stent deployment, enable performing a complete angioplasty procedure.ResultsWe present detailed results of simulation accuracy of the virtual instruments, along with their computational performance. In addition, the results of a preliminary face and content validation study conveyed on a group of 17 interventional radiologists are given.ConclusionsVR simulation of cardiovascular procedure can contribute to surgical training and improve the educational experience without putting patients at risk, raising ethical issues or requiring expensive animal or cadaver facilities. VCSim3 is still a prototype, yet the initial results indicate that it provides promising foundations for further development.

Real time simulation of inextensible surgical thread using a Kirchhoff rod model with force output for haptic feedback applications

We discuss the dynamics of an inextensible thin Kirchhoff rod used in the modeling of surgical threads, and demonstrate a very efficient scheme to not only simulate the motion of the thread in real-time (up to 1 ms per frame) but also obtain the constraint axial forces which can be fed back to a haptic system. The numerical scheme is based on a family of schemes called geometric or discrete variational integrators guaranteeing that the momentum and energy are exactly conserved over long periods of time for conservative systems. Besides, we report on an efficient numerical procedure to handle the inextensibility of the thread through physically based Lagrange multipliers, as well as the internal dissipation of the thread. We have performed simulations to verify the capabilities of our model to conserve momentum and energy, accurately calculate the axial constraint forces along the thread for haptic feedback, and capture bending-torsion coupling leading to the formation of plectonemes. While many of the ideas are well known in the computer graphics community (especially in hair modeling), we have implemented several improvements for the specific purpose of speeding up the computations for developing physically based haptic interfaces for knot tying and suturing.

Form-Finding of Interlaced Space Structures Using Constrained Nonlinear Optimization

The term Interlaced Space Structures (ISS) corresponds to a coupled system of naturally curved flexible panels/strips interlaced together according to a design pattern. The goal of the present study is to look for a physically-based and efficient form-finding procedure in order to interactively explore different interlaced morphologies with respect to the design parameters for structural design purposes. Each panel is considered as an in-extensible discrete kirchhoff rod and the rest shape of the coupled system rods is obtained via a constrained total energy minimization. The interlacing pattern is translated into a set of overlap order constraints and applied to the optimization problem. An implementation of the interior-point filter linesearch algorithm with the Quasi-Newton procedure has been employed to solve the constrained nonlinear optimization. The approach is then discussed and commented in detail through a number of form-finding case studies.

Surface Constrained Elastic Rods with Application to the Sphere

The unilateral contact between an elastic rod and a rigid surface is encountered in numerous biological and engineering applications. Along continuous contacts, the centerline of the rod reflects the geometry of the constraining surface. This restriction of the rod axis to surface bound configurations enters its local equilibrium through a reaction pressure which ensures the compatibility between the deformation and the restraint geometry. The classic equations governing the static equilibrium of elastic rods are particularized to surface bound configurations by (i) specifying the location of the rod axis in terms of its coordinates in the parameter space associated with the constraining surface parameterization, and (ii) characterizing the orientation of its material frame through its rotation with respect to the surface normal. This formulation, which emphasizes the relations between the rod configuration and the geometry of the constraint, also leads to an expression for the reaction pressure. This approach is then validated on a spherical surface by comparing with known solutions for elastic curves, i.e., inextensible and twist free rods.

Theoretical and experimental study of the stability of a soap film spanning a flexible loop

A variational approach is used to study the stability of a soap film spanning a flexible loop. The film is modeled as a fluid surface endowed with constant tension and the loop is modeled as an inextensible and unshearable elastic rod that resists not only bending but also twisting about the tangent to its centerline. The first and second variations of the underlying energy functional are derived, leading to governing equilibrium equations and energetically based stability conditions. The latter conditions are applied to explore the effect of subjecting flat circular configurations to planar and transverse perturbations. Both the areal stretching and the lineal twisting prove to compete against the lineal bending to destabilize such a configuration. Experiments are performed to study post-buckled configurations and to assess the validity of the theoretical predictions.

Non-periodic homogenization of bending–torsion theory for inextensible rods from 3D elasticity

We derive, by means of \(\Gamma \)-convergence, the equations of homogenized bending rod starting from 3D nonlinear elasticity equations. The main assumption is that the energy behaves like \(h^2\) (after dividing by \(h^2\), the order of vanishing volume), where h is the thickness of the body. We do not presuppose any kind of periodicity and work in the general framework. The result shows that, on a subsequence, we always obtain the equations of the same type as in bending–torsion rod theory and identifies, in an abstract formulation, the limiting quadratic form connected with that model. This result is the generalization of periodic homogenization of bending–torsion rod theory already present in the literature.

In-plane and out-of-plane free vibration and stability of a curved rod in flow

This paper investigates the free vibration and stability of a curved rod in flow. The equations of the three-dimensional motions of the rod are derived by the Newtonian approach. The differential quadrature method (DQM) is introduced to formulate the discrete forms of the governing equations of the inextensible rod with clamped–clamped supports. Based on numerical calculations, the effects of several system parameters, especially the flow velocity, on the natural frequencies and stability of the system are discussed. Buckling and flutter instability are detected as the flow velocity is varied in a certain range. Moreover, a derivation of the generalized slender-body theory for such a deformable curved rod is given in Appendix A.

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.

Instability of solitons under flexure and twist of an elastic rod

We study the stability of planar soliton solutions of equations describing the dynamics of an infinite inextensible unshearable rod under three-dimensional spatial perturbations. As a result of linearization about the soliton solution, we obtain an inhomogeneous scalar equation. This equation leads to a generalized eigenvalue problem. To establish the instability, we must verify the existence of an unstable eigenvalue (an eigenvalue with a positive real part). The corresponding proof of the instability is done using a local construction of the Evans function depending only on the spectral parameter. This function is analytic in the right half of the complex plane and has at least one zero on the positive real axis coinciding with an unstable eigenvalue of the generalized spectral problem.

Rigorous Derivation of a Homogenized Bending-Torsion Theory for Inextensible Rods from Three-Dimensional Elasticity

Rigorous Derivation of a Homogenized Bending-Torsion Theory for Inextensible Rods from Three-Dimensional Elasticity

Tensile behaviors of pre-twisted composite strands

In this paper, we address the tensile behaviors of pre-twisted composite strands, which consists of a pre-twisted single core filament surrounded by n-helical side filaments. Based on the extensible rod with zero bending and small twisting moduli for the core filament and inextensible rod for the side filaments, we develop the analytical method of the tensile behaviors of pre-twisted composite strands. Using a numerical continuation scheme, we elucidate the effects of microscopic factors such as initial helical angle, pre-twist of core filament, ratio of elastic modulus of core to that of side filament, and the number of the side filaments on the macroscopic tensile behavior of the strand as a whole structure. As a result, we show that the behavior is not trivial, even though the filament is a linear elastic due to the interplay of both geometrical constraint and finite deformation of the strand.

Stability Estimates for a Twisted Rod Under Terminal Loads: A Three-dimensional Study

The stability of an inextensible unshearable elastic rod with quadratic strain energy density subject to end loads is considered. We study the second variation of the corresponding rod-energy, making a distinction between in-plane and out-of-plane perturbations and isotropic and anisotropic cross-sections, respectively. In all cases, we demonstrate that the naturally straight state is a local energy minimizer in parameter regimes specified by material constants. These stability results are also accompanied by instability results in parameter regimes defined in terms of material constants.

On the curve straightening flow of inextensible, open, planar curves

We consider the curve straightening flow of inextensible, open, planar curves generated by the Kirchhoff bending energy. It can be considered as a model for the motion of elastic, inextensible rods in a high friction regime. We derive governing equations, namely a semilinear fourth order parabolic equation for the indicatrix and a second order elliptic equation for the Lagrange multiplier. We prove existence and regularity of solutions, compute the energy dissipation, prove its coercivity and conclude convergence to equilibrium, namely to a straight curve, at an exponential rate.

Dynamics of plane motion of a body with a system of flexible inextensible rods connected in series by elastoviscous joints at large angles of rotation

The principle of virtual displacements is used to obtain the equations of plane motion, in generalized coordinates, of a free rigid body with a system of flexible inextensible rods connected in series by elastoviscous joints at large angles of rotation. Each rod rotates as a line connecting its ends and bends according to two given shapes. The imposition of the kinematic conditions of inextensibility of the rods ensures that the mathematical model of the system does not contain oscillations caused by longitudinal vibrations of the rods. This fact improves the computational stability of the system.

Brownian dynamics simulations of charged semiflexible polymers confined to curved surfaces

As an extension of the generalized bead–rod model developed earlier by the authors, this paper proposes a method for Brownian dynamics simulations of charged semiflexible polymers confined to various curved surfaces such as spherical, cylindrical, ellipsoidal and toroidal. We model charged semiflexible polymers as discrete wormlike chains consisting of virtual beads connected by inextensible rods with length varying according to the characteristic radius of curvature of the confining surface. The long-range electrostatic interactions are incorporated via the Debye–Hueckel potential. The geometrical constraints associated with the inextensible rods are realized by the so-called linear constraint solver. For a semiflexible polymer chain confined to a spherical surface, an analytical expression for the winding number is obtained by using an existing exact closed-form solution of the mean-square end-to-end distance. The proposed simulation method is then validated against theoretical predictions for both charged and uncharged polymer chains under surface confinements.