Slender Elastic Rods Research Articles

The Design Space of Kirchhoff Rods

The Kirchhoff rod model describes the bending and twisting of slender elastic rods in three dimensions and has been widely studied to enable the prediction of how a rod will deform, given its geometry and boundary conditions. In this work, we study a number of inverse problems with the goal of computing the geometry of a straight rod that will automatically deform to match a curved target shape after attaching its endpoints to a support structure. Our solution lets us finely control the static equilibrium state of a rod by varying the cross-sectional profiles along its length. We also show that the set of physically realizable equilibrium states admits a concise geometric description in terms of linear line complexes, which leads to very efficient computational design algorithms. Implemented in an interactive software tool, they allow us to convert three-dimensional hand-drawn spline curves to elastic rods and give feedback about the feasibility and practicality of a design in real time. We demonstrate the efficacy of our method by designing and manufacturing several physical prototypes with applications to interior design and soft robotics.

Automated Stability Testing of Elastic Rods With Helical Centerlines Using a Robotic System

Experimental analysis of the mechanics of a deformable object, and particularly its stability, requires repetitive testing and, depending on the complexity of the object’s shape, a testing setup that can manipulate many degrees of freedom at the object’s boundary. Motivated by recent advancements in robotic manipulation of deformable objects, this letter addresses these challenges by constructing a method for automated stability testing of a slender elastic rod — a canonical example of a deformable object — using a robotic system. We focus on rod configurations with helical centerlines since the stability of a helical rod can be described using only three parameters, but experimentally determining the stability requires manipulation of both the position and orientation at one end of the rod, which is not possible using traditional experimental methods that only actuate a limited number of degrees of freedom. Using a recent geometric characterization of stability for helical rods, we construct and implement a manipulation scheme to explore the space of stable helices, and we use a vision system to detect the onset of instabilities within this space. The experimental results obtained by our automated testing system show good agreement with numerical simulations of elastic rods in helical configurations. The methods described in this letter lay the groundwork for automation to grow within the field of experimental mechanics.

Writhing and hockling instabilities in twisted elastic fibers.

The buckling and twisting of slender, elastic fibers is a deep and well-studied field. A slender elastic rod that is twisted with respect to a fixed end will spontaneously form a loop, or hockle, to relieve the torsional stress that builds. Further twisting results in the formation of plectonemes-a helical excursion in the fiber that extends with additional twisting. Here we use an idealized, micron-scale experiment to investigate the energy stored, and subsequently released, by hockles and plectonemes as they are pulled apart, in analogy with force spectroscopy studies of DNA and protein folding. Hysteresis loops in the snapping and unsnapping inform the stored energy in the twisted fiber structures.

Three-dimensional nonlinear dynamics of prestressed active filaments: Flapping, swirling, and flipping.

Initially straight slender elastic filaments or rods with constrained ends buckle and form stable two-dimensional shapes when prestressed by bringing the ends together. Beyond a critical value of this prestress, rods can also deform off plane and form twisted three-dimensional equilibrium shapes. Here, we analyze the three-dimensional instabilities and dynamics of such deformed filaments subject to nonconservative active follower forces and fluid drag. We find that softly constrained filaments that are clamped at one end and pinned at the other exhibit stable two-dimensional planar flapping oscillations when active forces are directed toward the clamped end. Reversing the directionality of the forces quenches the instability. For strongly constrained filaments with both ends clamped, computations reveal an instability arising from the twist-bend-activity coupling. Planar oscillations are destabilized by off-planar perturbations resulting in twisted three-dimensional swirling patterns interspersed with periodic flipping or reversal of the swirling direction. These striking swirl-flip transitions are characterized by two distinct timescales: the time period for a swirl (rotation) and the time between flipping events. We interpret these reversals as relaxation oscillation events driven by accumulation of torsional energy. Each cycle is initiated by a fast jump in torsional deformation with a subsequent slow decrease in net torsion until the next cycle. Our work reveals the rich tapestry of spatiotemporal patterns when weakly inertial strongly damped rods are deformed by nonconservative active forces. Taken together, our results suggest avenues by which prestress, elasticity, and activity may be used to design synthetic macroscale pumps or mixers.

A Semi-Analytic Elastic Rod Model of Pediatric Spinal Deformity.

The mechanism of the scoliotic curve development in healthy adolescents remains unknown in the field of orthopedic surgery. Variations in the sagittal curvature of the spine are believed to be a leading cause of scoliosis in this patient population. Here, we formulate the mechanics of S-shaped slender elastic rods as a model for pediatric spine under physiological loading. Second, applying inverse mechanics to clinical data of the subtypes of scoliotic spines, with characteristic 3D deformity, we determine the undeformed geometry of the spine before the induction of scoliosis. Our result successfully reproduces the clinical data of the deformed spine under varying loads, confirming that the prescoliotic sagittal curvature of the spine impacts the 3D loading that leads to scoliosis.

3D Deformation Patterns of S Shaped Elastic Rods as a Pathogenesis Model for Spinal Deformity in Adolescent Idiopathic Scoliosis

Adolescent idiopathic scoliosis (AIS) is a three-dimensional (3D) deformity of the spinal column in pediatric population. The primary cause of scoliosis remains unknown. The lack of such understanding has hampered development of effective preventive methods for management of this disease. A long-held assumption in pathogenesis of AIS is that the upright spine in human plays an important role in induction of scoliosis. Here, the variations in the sagittal curve of the scoliotic and non-scoliotic pediatric spines were used to study whether specific sagittal curves, under physiological loadings, are prone to 3D deformation leading to scoliosis. To this end, finite element models of the S shaped elastic rods, which their curves were derived from the radiographs of 129 sagittal spinal curves of adolescents with and without scoliosis, were generated. Using the mechanics of deformation in elastic rods, this study showed that the 3D deformation patterns of the two-dimensional S shaped slender elastic rods mimics the 3D patterns of the spinal deformity in AIS patients with the same S shaped sagittal spinal curve. On the other hand, the rods representing the non-scoliotic sagittal spinal curves, under the same mechanical loading, did not twist thus did not lead to a 3D deformation. This study provided strong evidence that the shape of the sagittal profile in individuals can be a leading cause of the 3D spinal deformity as is observed in the AIS population.

Patterns of Carbon Nanotubes by Flow-Directed Deposition on Substrates with Architectured Topographies.

We develop and perform continuum mechanics simulations of carbon nanotube (CNT) deployment directed by a combination of surface topography and rarefied gas flow. We employ the discrete elastic rods method to model the deposition of CNT as a slender elastic rod that evolves in time under two external forces, namely, van der Waals (vdW) and aerodynamic drag. Our results confirm that this self-assembly process is analogous to a previously studied macroscopic system, the "elastic sewing machine", where an elastic rod deployed onto a moving substrate forms nonlinear patterns. In the case of CNTs, the complex patterns observed on the substrate, such as coils and serpentines, result from an intricate interplay between van der Waals attraction, rarefied aerodynamics, and elastic bending. We systematically sweep through the multidimensional parameter space to quantify the pattern morphology as a function of the relevant material, flow, and geometric parameters. Our findings are in good agreement with available experimental data. Scaling analysis involving the relevant forces helps rationalize our observations.

Large deflections of a clamped-slider-pinned rod with uniform intrinsic curvature

This paper examines large planar deflections of a slender elastic rod with uniform intrinsic curvature, which is clamped at one end and the other end is pinned to a slider allowing it to move freely in the vertical direction as the ends are displaced horizontally in a straight line. The analysis encompasses force-displacement loading paths and corresponding configurations, with a particular focus on the formation of loops. The analysis is accompanied with data obtained from experiments on nickel titanium alloy strips, demonstrating a good match with theoretical predictions.

Influence of the connecting condition on the dynamic buckling of longitudinal impact for an elastic rod

The stress wave propagation law and dynamic buckling critical velocity are formulated and solved by considering a general axial connecting boundary for a slender elastic straight rod impacted by a rigid body. The influence of connecting stiffness on the critical velocity is investigated with varied impactor mass and buckling time. The influences of rod length and rod mass on the critical velocity are also discussed. It is found that greater connecting stiffness leads to larger stress amplitude, and further results in lower critical velocity. It is particularly noteworthy that when the connecting stiffness is less than a certain value, dynamic buckling only occurs before stress wave reflects off the connecting end. It is also shown that longer rod with larger slenderness ratio is easier to buckle, and the critical velocity for a larger-mass rod is higher than that for a lighter rod with the same geometry.

Axisymmetric viscoelastic response of flexible pipes in time domain

The challenges for determining the mechanical behavior of flexible pipes mainly arise from highly non-linear geometrical and material properties and complex contact interaction conditions between and within layers components. This paper develops an innovative model to investigate the linear viscoelastic behavior of flexible pipes under axisymmetric loads in time domain. The model is derived from an equivalent linear elastic axisymmetric model by invoking the elastic-viscoelastic correspondence principle. Analytical formulations that describe the behavior of the metallic helical layers based on a combination of differential geometry concepts and Clebsch–Kirchhoff equilibrium equations for initially curved slender elastic rods are presented. The elastic response of the homogenous polymeric cylindrical layers is also presented. The assemblage of both types of governing algebraic equations that approximate analytical solutions for force and moment distributions, deformations in each layer, as well as contact pressure between near layers, taking time-dependent characteristics of polymeric layers into account are provided and it is clear that the relationship between axial force and elongation is non-linear and encompasses a hysteretic response. Besides, the creep behavior in axial direction can also be found. Some insights into the differences in the behavior for several loading conditions are discussed by considering variable frequencies.

A Geometric Model for the Coiling of an Elastic Rod Deployed Onto a Moving Substrate

We report results from a systematic numerical investigation of the nonlinear patterns that emerge when a slender elastic rod is deployed onto a moving substrate; a system also known as the elastic sewing machine (ESM). The discrete elastic rods (DER) method is employed to quantitatively characterize the coiling patterns, and a comprehensive classification scheme is introduced based on their Fourier spectrum. Our analysis yields physical insight on both the length scales excited by the ESM, as well as the morphology of the patterns. The coiling process is then rationalized using a reduced geometric model (GM) for the evolution of the position and orientation of the contact point between the rod and the belt, as well as the curvature of the rod near contact. This geometric description reproduces almost all of the coiling patterns of the ESM and allows us to establish a unifying bridge between our elastic problem and the analogous patterns obtained when depositing a viscous thread onto a moving surface; a well-known system known as the fluid-mechanical sewing machine (FMSM).

Untangling the mechanics and topology in the frictional response of long overhand elastic knots.

We combine experiments and theory to study the mechanics of overhand knots in slender elastic rods under tension. The equilibrium shape of the knot is governed by an interplay between topology, friction, and bending. We use precision model experiments to quantify the dependence of the mechanical response of the knot as a function of the geometry of the self-contacting region, and for different topologies as measured by their crossing number. An analytical model based on the nonlinear theory of thin elastic rods is then developed to describe how the physical and topological parameters of the knot set the tensile force required for equilibrium. Excellent agreement is found between theory and experiments for overhand knots over a wide range of crossing numbers.

ON MOTION OF AN ELASTIC WIRE IN A RIEMANNIAN MANIFOLD AND SINGULAR PERTURBATION

R.E. Caflish and J.H. Maddocks analyzed the dynamics of a planar slender elastic rod. We consider a thin elastic rod $\gamma$ in an $N$-dimensional riemannian manifold. The former model represents an elastic rod with positive thickness, and the equation becomes a semilinear wave equation. Our model represents an infinitely thin elastic rod, and the equation becomes a 1-dimensional semilinear plate equation. We prove the short time existence of solutions. We also discuss the behaviour of the solution when the resistance goes to infinity, and find that the solution converges to a solution of a gradient flow equation.

Pattern morphology in the elastic sewing machine

We present results from a numerical investigation of the coiling patterns obtained when a slender elastic rod is deployed onto a moving substrate. The Discrete Elastic Rods method is employed to explore the parameter space, construct phase diagrams, identify their phase boundaries and characterize the pattern morphology. The various length scales of the patterns are primarily set by the gravity-bending length and depend only logarithmically on the deployment height. The curvature near the contact point, together with the dimensionless speed mismatch between deployment and the belt, dictate the characteristics of the patterns. The phase boundaries are found to be independent of both the gravito-bending length and the deployment height, as long as the latter is above a threshold value. We also evaluate the relative importance of twist and curvature strains, which confirms that bending has a dominant role.

Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method

We present a theoretical and numerical framework to compute bifurcations of equilibria and stability of slender elastic rods. 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 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 method gives interactive access to semi-analytical equilibrium branches, in contrast with the individual solution points obtained from classical minimization or predictor–corrector techniques. By way of example, we apply our numerics to address the specific problem of a naturally curved and heavy rod under extreme twisting and perform a detailed comparison against our own precision model experiments of this system. Excellent quantitative agreement is found between experiments and simulations for the underlying 3D buckling instabilities and the characterization of the resulting complex configurations. We believe that our framework is a powerful alternative to other methods for the computation of nonlinear equilibrium 3D shapes of rods in practical scenarios.

Geometrically non-linear analysis of inclined elastic rods subjected to self-weight

The behavior of inclined slender elastic rods subjected to axial forces and distributed load is discussed in this paper. Mathematical models and numerical solutions are developed for small and large displacements. A double-hinged boundary condition is assumed and the analysis is carried out for different values of non-dimensional weight (distributed load) and angle of inclination. The mathematical formulation results from considering geometrical compatibility, equilibrium of forces and moments and constitutive relations. For large displacements, a set of six first order non-linear ordinary differential equations with boundary conditions prescribed at both ends is obtained. This two-point boundary value problem is numerically integrated using a three-parameter shooting method. When small displacements are assumed the problem simplifies and a power series solution may be conveniently employed. The results for both simulations are presented, compared and discussed.

Shear flow past slender elastic rods attached to a plane

Shear flow past an elastic rod or a doubly periodic array of elastic rods attached to a plane is investigated with reference to flow over a ciliated surface or a carbon nanomat. In the absence of flow, the rods are straight cylinders with circular cross-sectional shape. The mathematical framework combines slender-body theory for computing the hydrodynamic load with classical beam theory for computing the rod deflection. Small deflections are computed using a finite-element method and large deflections are computed using an iterative procedure where a rod shape is assumed, the hydrodynamic load is found, and a new shape is obtained by solving a boundary-value problem involving ordinary differential equations at equilibrium. Deflected rod profiles are presented and the macroscopic slip velocity is computed in the case of flow past a doubly periodic square or triangular array of rods over a broad range of surface coverage.

Kink Instability of a Highly Deformable Elastic Cylinder

When a soft elastic cylinder is bent beyond a critical radius of curvature, a sharp fold in the form of a kink appears catastrophically at its compressed side while the tensile side remains smooth. The critical radius increases linearly with the diameter of the cylinder but remains independent of its material properties such as modulus; the maximum deflection at the location of the kink depends on both the material and geometric properties of the cylinder. The catastrophic dynamics of evolution of the kink depicts propagation of a shear wave from the location of the kink towards the edges signifying that kinking is an elastic response of the material which results in extreme localization of curvature. We have rationalized this phenomenon in the light of the classical Euler's buckling instability in slender elastic rods.

Initial Postbuckling of Elastic Rods Subjected to Thermal Loads and Resting on an Elastic Foundation

In practical applications encountered in some mechanical systems initially straight slender elements laterally supported by elastic foundations may be prevented from thermal expansion. In such conditions a compressive force is created and a serious equilibrium instability phenomenon may take place when a critical temperature gradient is imparted to the structure. If the temperature is further increased the rod may experience relatively large deflections, and a geometrically non-linear two-point boundary value problem describes this behaviour. However, if finite but moderate deflections are admitted, an approximate solution may be obtained if a classical perturbation expansion method is applied to the non-linear ordinary differential governing equations. This approach renders a set of linear equations which can be sequentially solved. This work proposes an analytical solution for the assessment of the initial post-buckling configuration of slender elastic rods supported on linear elastic foundations and subjected to a uniform temperature gradient. The rod ends are assumed hinged and immovable and the thermal strain-temperature relationship is linear. The governing equations are derived and made non-dimensional so it is seen that two parameters rule the problem: the foundation stiffness modulus and the rod slenderness ratio. Results are presented and discussed for one slenderness ratio and a range of foundation stiffness that corresponds to the fourth buckling mode. Generally an increase in the applied temperature is associated with a decrease in the compressive force, but it is shown particular conditions where an opposite behavior is observed. The initial post-buckling results for unsupported rods are compared with existing full post-buckling solution and show excellent agreement, given that in practical situations the deflections are moderate.

Post-buckling analysis of slender elastic vertical rods subjected to terminal forces and self-weight

This article presents the behavior of slender elastic rods subjected to axial terminal forces and self-weight. The mathematical formulation is presented, a solution is sought for a double-hinged boundary condition and the analysis is carried out for different values of non-dimensional weight. The formulation derives from geometrical compatibility, equilibrium of forces and moments and constitutive relations yielding a set of six first order non-linear ordinary differential equations with boundary conditions specified at both ends, which characterizes a complex two-point boundary value problem. Furthermore, a perturbation method is used to find the critical buckling loads and initial post-buckling solutions. A numerical integration scheme based on a three parameter shooting method is employed in the post-buckling solutions.