Elastic Rod Research Articles

Buckling by disordered growth

Buckling instabilities driven by tissue growth underpin key developmental events such as the folding of the brain. Tissue growth is disordered due to cell-to-cell variability, but the effects of this variability on buckling are unknown. Here, we analyze what is perhaps the simplest setup of this problem: the buckling of an elastic rod with fixed ends driven by spatially varying, yet highly symmetric growth. Combining analytical calculations for simple growth fields and numerical sampling of random growth fields, we show that variability can increase as well as decrease the growth threshold for buckling, even when growth variability does not cause any residual stresses. For random growth, we find numerically that the shift of the buckling threshold correlates with spatial moments of the growth field. Our results imply that biological systems can either trigger or avoid buckling by exploiting the spatial arrangement of growth variability. Published by the American Physical Society 2024

A model for micro-scale propulsion using flexible rotating flagella

Micro-scale propulsion by rotating helical flagella is of interest for the study of bacteria and robotic micro-swimmers. The propulsive thrust and torque produced by the rotating flagella are usually estimated assuming that they are rigid. In this paper we assume the flagella to be deformable elastic rods and compute propulsive forces and torques by enforcing local equilibrium of the rod within the context of resistive force theory. The torque-speed characteristics of the flagellar motor driving the rotation are taken into account. We show that the problem can be cast as a system of algebraic equations if the flagella are assumed to be helical before and after deformation when no spontaneous curvature is included. If the assumption of helical shape is dropped then we show that the propulsion problem can be cast as a system of first order differential equations that can be solved numerically. Our results in both cases agree reasonably well with experimental observations of bacterial propulsion and deviate from the predictions of Purcell depending on the mechanical properties of the flagellum.

Effective extensional–torsional elasticity and dynamics of helical filaments under distributed loads

We study slender, helical elastic rods subject to distributed forces and moments. Focussing on the case when the helix axis remains straight, we employ the method of multiple scales to systematically derive an ‘equivalent-rod’ theory from the Kirchhoff rod equations: the helical filament is described as a naturally-straight rod (aligned with the helix axis) for which the extensional and torsional deformations are coupled. Importantly, our analysis is asymptotically exact in the limit of a ‘highly-coiled’ filament (i.e., when the helical wavelength is much smaller than the characteristic lengthscale over which the filament bends due to external loading) and is able to account for large, unsteady displacements. In addition, our analysis yields explicit conditions on the external loading that must be satisfied for a straight helix axis. In the small-deformation limit, we exactly recover the coupled wave equations used to describe the free vibrations of helical coil springs, thereby justifying previous equivalent-rod approximations in which linearised stiffness coefficients are assumed to apply locally and dynamically. We then illustrate our theory with two loading scenarios: (I) a heavy helical rod deforming under its own weight; and (II) the dynamics of axial rotation (twirling) in viscous fluid, which may be considered as a simple model for a bacteria flagellar filament. In both scenarios, we demonstrate excellent agreement with solutions of the full Kirchhoff rod equations, even beyond the formal limit of validity of the ‘highly-coiled’ assumption. More broadly, our analysis provides a framework to develop reduced models of helical rods in a wide variety of physical and biological settings, and yields analytical insight into their elastic instabilities. In particular, our analysis indicates that tensile instabilities are a generic phenomenon when helical rods are subject to a combination of distributed forces and moments.

СЕМАНТИЧЕСКАЯ БЛИЗОСТЬ АСТРОНОМИЧЕСКИХ ПОНЯТИЙ, ОБОЗНАЧАЮЩИХ НЕБЕСНЫЕ ТЕЛА

The problem of determining the semantic similarity of astronomical terms denoting celestial objects in a school astronomy course is considered. A distributive method is used, which consists in comparing the distributions of concepts in the astronomy textbook. With the help of special computer programs, punctuation marks and stop words are removed from the text, the number of mentions of various terms is deter-mined, the cosine measure of proximity and the semantic distance between concepts are calculated. This allows: 1) to reveal frequently used astronomical concepts denoting celestial bodies; 2) to obtain matrices of proximity and semantic distances between the concepts of "galaxy", "star", "Sun", "planet", "black hole", "white dwarf", etc.; 3) to create a mental map of the celestial bodies that make up the Solar system, taking into account the distances between the terms designating them 4) to build clouds of concepts "types of stars", "Earth – star – Universe". To model the semantic space of the textbook, the Graf.pas computer program is used, which randomly moves vertices in 2D or 3D space, each time calculating the potential energy W of the elastic rods connecting them and choosing such an arrangement of vertices that W is minimal. It is believed that the closely located vertices are connected by rigid rods, and the far removed ones are connected by elastic rubber threads.

Forced vibrations of the shaft line based on the model of a two-stage rod with an elastic connection of sections

The paper considers a method for solving the problem of dynamics of forced vibrations of a wall line based on a model of an equivalent elastic two-stage rod with a pliable connection of sections. The model is represented by a system of partial differential equations. The solution of the equations is obtained by the Fourier method for eigenfunctions orthogonal with weight. It is established that the presence of an elastic element at the junction of the sections does not affect the orthogonality of the eigenfunctions. The influence of the elastic coupling compliance coefficient on the natural frequency spectrum of the shaft line is estimated. The obtained forms of natural vibrations of an equivalent elastic two-stage rod are consistent in appearance with the corresponding forms known in the literature for a discrete model of a similar torsional circuit. A partial solution of the equations is found for the case of the action of the harmonic load on the engine and the averaged torque on the propeller. As an example, the dynamics of forced vibrations of the shaft line of a barge of the Sosnovka type is studied. The results of calculating the natural frequencies of the considered model are compared with the classical discrete mass method. The analysis of the obtained plots of the twisting angles and torques showed that the angles are satisfactorily approximated by the first proper oscillation shape, and for moments, high shapes must be taken into account. The comparison of the simulation results with the data of the torsiogram obtained under production testing conditions indicates the need for further development of the model. Further improvement of the torsional circuit, as well as a more detailed study of the nature of the operating loads, are considered as promising areas of research.

Efficient Uncertainty Quantification for Mechanical Properties of Randomly Perturbed Elastic Rods

Efficient Uncertainty Quantification for Mechanical Properties of Randomly Perturbed Elastic Rods

СТІЙКІСТЬ ЗАЛІЗОБЕТОННИХ СТІЙОК ПРИ КОРОТКОЧАСНІЙ ТА ТРИВАЛІЙ ДІЇ НАВАНТАЖЕННЯ

The behavior of hinged support at the ends of the rack with initial bending under the action of load is considered. Spring rods are arranged symmetrically. The rack has an initial bend, compressed by forces constant in time. All real elements have one or another imperfections in the form of technological bends, therefore, they begin to bend from the very beginning of the load. The load, when it is exceeded even by an infinitesimally small amount, there is a loss of stability of this type of deformation, is called critical. In the calculation of stability under the long-term action of external forces, it is necessary to determine the load, at which the rate of movement in time monotonically decays. Solving the problem in such a setting is acceptable for systems, the development of movements of which in time leads to a change in the stress state. This condition for a compressed rack is fulfilled only in the presence of initial imperfections (initial bending, off-center application of compressive force, etc.). When solving the problems of the theory of stability, taking into account the creep of the material plays an important role. Creep can be limited in time or unlimited. When solving the problem of the stability of racks with initial imperfections, made of a material that has creep and reinforced with elastic rods, the following assumptions are made: 1) the hypothesis of flat sections is considered valid; 2) the deformations of the creeping material and the elastic rods at the points of contact are the same; 3) the modulus of deformations during stretching and compression are equal; 4) the creep material works in the stretched zone without the appearance of cracks. The relationship between deformations and elasticities in the material of the rack is established by a formula based on the linear relationship between deformations and elasticities. The creep rate of concrete is based on the hereditary theory of aging. In the work, an integro-differential equation was obtained - the equation of slow motion of the rod, expressions were obtained for the study of bends in any at what point in time, the formula for determining the critical force with long-term load action is derived.

Free-falling motion of an elastic rigid rod towards a Schwarzschild black hole

Abstract We study the motion of an elastic rigid rod which is radially free-falling towards a Schwarzschild black hole. This is accomplished by reducing the corresponding free-boundary partial differential equation problem to a sequence of ODEs, which we integrate numerically. Starting with a rod at rest, we show that it is possible to choose its initial compression profile so that its midpoint falls substantially faster, or slower, than a free-falling particle with the same initial conditions. This seems to be a purely kinematic effect, since on average there is no net transfer of elastic energy to mechanical energy.

Simultaneous position and force control of a SMA-actuated continuum robotic module

Continuum manipulators with controllable shape and exerting force are attractive devices. Utilizing great features of shape memory alloys (SMAs), thin continuum modules are developed. SMAs are embedded around an elastic rod and activated through a control algorithm to set a desired shape. However, complex behavior of SMAs in such a multi-input multi-output system, make the control challenging. Additionally, simultaneous control of position and the force applied by the module, as a challenging problem, has not been investigated so far. Handling this problem may increase the application of continuum robots when utilized as manipulators or when pass through narrow and complex canals with sensitive wall. In this research, position and force control of such continuum module is under focus for achieving a more practical tool for better non-invasive medical devices. Further to the position control, the amount of force applied to the environment is adjusted in different locations of the workspace through a novel fuzzy controller. The results indicate the possibility of simultaneous control of the position and force using fuzzy controller with a reasonable accuracy of 0.5° for angle and 0.05 N for the force. The concluded results may be utilized in developing smarter soft robots.

Anisotropic Stick-Slip Frictional Surfaces via Titania Nanorod Patterning.

Nanoscale or microscale surface texturing is an effective technique to tailor the tribological properties between two surfaces that are rubbed against each other. In order to achieve the desired frictional properties by a patterned surface, one needs an in-depth understanding of the underlying mechanisms. Here, we demonstrate anisotropic stick-slip friction achieved via a nanotextured surface of tilted titania nanorods (TiNRs). The surface was developed by using the glancing angle deposition (GLAD) technique, and exhibited load-dependent variations in stick-slip friction as well as frictional anisotropy in different sliding directions. For studying the frictional properties of the newly developed surface, lateral force microscopy (LFM) was performed in three different reciprocal orientations (0° rotated, 45° rotated, 90° rotated) using a custom-made colloidal alumina atomic force microscopy (AFM) probe. The frictional behavior was found to vary significantly with the orientation. At 0° rotated position) a prominent "stick-slip" was observed when scanning opposite to the tilt direction, whereas the phenomenon reduced significantly when the nanotextured surface was scanned along the tilt direction or rotated to different angles (45 and 90°) with respect to the sliding direction of the AFM cantilever supporting the probe. The experimental findings were interpreted based on the classical solution for large deflections of tilted elastic rods. Overall, the textured surface, LFM-based frictional measurement, and the quantitative analysis presented here provide a fundamental understanding of how friction can be significantly varied on a surface patterned with tilted TiNRs at a length scale of about 1 μm, which can be comprehensively applied to nanorod patterns of other materials on different substrates.

A comprehensive numerical approach to coil placement in cerebral aneurysms: mathematical modeling and in silico occlusion classification.

Endovascular coil embolization is one of the primary treatment techniques for cerebral aneurysms. Although it is a well-established and minimally invasive method, it bears the risk of suboptimal coil placement which can lead to incomplete occlusion of the aneurysm possibly causing recurrence. One of the key features of coils is that they have an imprinted natural shape supporting the fixation within the aneurysm. For the spatial discretization, our mathematical coil model is based on the discrete elastic rod model which results in a dimension-reduced 1D system of differential equations. We include bending and twisting responses to account for the coils natural curvature and allow for the placement of several coils having different material parameters. Collisions between coil segments and the aneurysm wall are handled by an efficient contact algorithm that relies on an octree based collision detection. In time, we use a standard symplectic semi-implicit Euler time stepping method. Our model can be easily incorporated into blood flow simulations of embolized aneurysms. In order to differentiate optimal from suboptimal placements, we employ a suitable in silico Raymond-Roy-type occlusion classification and measure the local packing density in the aneurysm at its neck, wall region and core. We investigate the impact of uncertainties in the coil parameters and embolization procedure. To this end, we vary the position and the angle of insertion of the micro-catheter, and approximate the local packing density distributions by evaluating sample statistics.

Analysis of the periodicity of movements of mechanisms with elastic links of variable length

Abstract Elastic dynamic systems with open kinematic chains are effectively described by a recursive model of the generalized Newton–Euler method with the provision of computational algorithms with a degree of complexity proportional to the dimension of these systems, i.e. O(n). Such systems, in particular, are elastic manipulators, some of which are discussed in this article. In the case when elastic dynamical systems have closed kinematic circuits, the application of a numerical analysis strategy without reversing the mass matrix is extremely difficult. The analysis of the simplest type of elastic mechanisms with closed circuits, such as a crank–slide mechanism and a four–pin mechanism with elastic connecting rods, revealed a peculiar picture of the periodicity of the functions of elastic movements of connecting rods, which differs significantly from the periods of operation of the mechanisms themselves. In this article, the properties of the periodicity (or pseudo-periodicity) of the elastic displacement function of mechanisms with elastic links of variable length are investigated using examples of numerical dynamic analysis of crank–rocker mechanisms.

High accuracy solutions for the Pochhammer–Chree equation in elastic media

This study investigates the nonlinear Pochhammer–Chree equation, a model crucial for understanding wave propagation in elastic rods, through the application of the Khater III method. The research aims to derive precise analytical solutions and validate them using He’s variational iteration method (VIM). The Pochhammer–Chree equation’s relationship to other nonlinear evolution equations, such as the Korteweg-de Vries and nonlinear Schrödinger equations, underscores its significance in the field of nonlinear wave dynamics. The methodology employs the Khater III method for deriving analytical solutions, while He’s VIM serves as a numerical validation tool, ensuring the accuracy and stability of the obtained results. This dual approach not only yields novel solutions but also provides a robust framework for analyzing complex wave phenomena in elastic media. The findings of this study have significant implications for material science and engineering applications, offering new insights into the behavior of waves in elastic rods. By bridging the gap between theoretical models and practical applications, this research contributes to the advancement of both mathematical theory and physical understanding of nonlinear wave dynamics. Situated within the domain of applied mathematics, with a focus on nonlinear wave equations, this work exemplifies the interdisciplinary nature of contemporary research in mathematical physics. The results presented herein open new avenues for future investigations in related fields and highlight the potential for innovative applications in material science and engineering.

Contact detection between curved fibres: high order makes a difference

Computer Graphics has a long history in the design of effective algorithms for handling contact and friction between solid objects. For the sake of simplicity and versatility, most methods rely on low-order primitives such as line segments or triangles, both for the detection and the response stages. In this paper we carefully analyse, in the case of fibre systems, the impact of such choices on the retrieved contact forces. We highlight the presence of artifacts in the force response that are tightly related to the low-order geometry used for contact detection. Our analysis draws upon thorough comparisons between the high-order super-helix model and the low-order discrete elastic rod model. These reveal that when coupled to a low-order, segment-based detection scheme, both models yield spurious jumps in the contact force profile. Moreover, these artifacts are shown to be all the more visible as the geometry of fibres at contact is curved. In order to remove such artifacts we develop an accurate high-order detection scheme between two smooth curves, which relies on an efficient adaptive pruning strategy. We use this algorithm to detect contact between super-helices at high precision, allowing us to recover, in the range of wavy to highly curly fibres, much smoother force profiles during sliding motion than with a classical segment-based strategy. Furthermore, we show that our approach offers better scaling properties in terms of efficiency vs. precision compared to segment-based approaches, making it attractive for applications where accurate and reliable forces are desired. Finally, we demonstrate the robustness and accuracy of our fully high-order approach on a challenging hair combing scenario.

Classical solution to mixed problems from the theory of longitudinal impact on an elastic semi-infinite rod in the case of separation of the impacting body after the collision

In this work, we consider two coupled initial-boundary value problems, which, based on the Saint-Venant theory, model the longitudinal impact phenomena in a semi-infinite rod. The mathematical formulation of the problem is two mixed problems for the one-dimensional wave equation with conjugation conditions. The Cauchy conditions are specified on the spatial half-line. The initial condition for the partial derivative with respect to the time variable has a discontinuity of the first kind at one point. The boundary condition, which includes the unknown function and its first- and second-order partial derivatives, is specified on the time half-line. The solution is constructed by the method of characteristics in an explicit analytical form. The uniqueness of the solution is proved, and the conditions under which a piecewise-smooth solution exists are established. The classical solution to a mixed problem with matching conditions is considered.

Deformation, shape transformations, and stability of elastic rod loops within spherical confinement

Mechanical insight into the packing of slender objects within confinement is essential for understanding how polymers, filaments, or wires organize and rearrange in limited space. Here we combine theoretical modeling, numerical optimization, and experimental studies to reveal spherical packing behavior of thin elastic rod loops of homogeneous or inhomogeneous stiffness. Across varying loop lengths, a rich array of configurations including circle, saddle, figure-eight, and more intricate patterns are identified. A theoretical framework rooted in the local equilibrium of force and moment is proposed for the rod loop deformation, facilitating the determination of internal and contact forces experienced by the rods during deformation. For the confined homogeneous rod loops, their stable and metastable configurations are well described using proposed Euler rotation curves, which offer a concise and effective approach for configuration prediction. Moreover, formulated analysis on the stability and critical force for homogeneous rod loops on great circles of the spherical confinement are performed. For inhomogeneous rod loops with two segments of differing stiffness, the stiffer segment exhibits less deviation from the great circle, while the softer segment undergoes more pronounced deformation. These findings not only enhance our understanding of buckling and post-buckling phenomena but also offer insights into filament patterning within confining environments.

MERCI: Mixed curvature-based elements for computing equilibria of thin elastic ribbons

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 growing number of applications in Computer Design. In this paper, starting from the reduced developable ribbon models [Sadowsky 1929; Wunderlich 1962] recently popularised in Soft Matter Physics, we propose a both accurate and efficient algorithm for computing the statics of thin elastic ribbons. Inspired by the super-clothoid model for thin elastic rods, our method relies on compact ribbon elements whose normal curvature varies linearly with respect to arc length s , while their geodesic torsion is quadratic in s . In contrast however, for the sake of efficiency, our algorithm avoids building a fully reduced kinematic chain and instead treats each element independently, gluing them only at the final solving stage through well-chosen bilateral constraints. Thanks to this mixed variational strategy, which yields a banded Hessian, our algorithm recovers the linear complexity of low-order models while preserving the quadratic convergence of curvature-based models. As a result, our approach is scalable to a large number of elements, and suitable for various boundary conditions and unilateral contact constraints, making it possible to handle challenging scenarios such as confined buckling experiments or Möbius bands with contact. Remarkably, our mixed algorithm proves an order of magnitude faster compared to Discrete Elastic Ribbon models of the literature while achieving, in a few seconds only, high accuracy levels that remain out of reach for such low-order models. Additionally, our numerical model can incorporate various ribbon energies, including the RibExt model for quasi-developable ribbons recently introduced in Physics [Audoly and Neukirch 2021], which allows to transition smoothly between a rectangular Kirchhoff rod and a (developable) Sadowsky ribbon. Our numerical scheme is carefully validated against demanding experiments of the Physics literature, which demonstrates its accuracy, efficiency, robustness, and versatility. Our Merci code is publicly available at https://gitlab.inria.fr/elan-public-code/merci for the sake of reproducibility and future benchmarking.

Перелом шийки променевої кістки в дітей: клініка, діагностика, методи лікування

Radial neck fractures are most common in children aged 8 to 11 years, accounting for 5% to 10% of all fractures of the bones that make up the elbow joint and up to 1% of all fractures in the pediatric population. Most often, the injury occurs as a result of a fall on an outstretched arm in the position of extension in the elbow joint. indirect injury from the fall with emphasis on the arm straightened at the elbow joint with the forearm deflected outward, and less often due to the direct effect of a damaging effect. The most typical damage in the area of the proximal radius in children is transverse fractures of the neck, in which the plane of the fracture passes through its metaphysis. The aim is to compare the experience of treating fractures of the radial neck, to analyze the possibilities of postoperative complications based on the literature and our own observations. In children with radial neck fractures, depending on the type of displacement of bone fragments and their angulation, there are different options for repositioning, which, accordingly, provide different treatment efficiency. Treatment of radial neck fractures in the pediatric population, necessarily taking into account the type of fragment displacement, should involve starting with low-traumatic and minimally invasive closed methods, and only in cases of their failure and/or impossibility (inappropriateness) it is possible to determine the choice of indications for open surgical intervention. The results which we obtained from the application of the open reposition technique for radial neck fractures according to the Metaizo method with fixation of fragments with elastic intramedullary rods in childhood clinically, radiologically and functionally coincide with the existing modern data of literary sources. Based on this, we believe that this technique is a good option for surgical treatment of pathology, simple and reliable, with minimal possibilities of postoperative complications, even with the most significant values of angulation of bone fragments. The research was carried out in accordance with the principles of the Declaration of Helsinki. The research protocol was approved by the Local Ethics Committee of the institution mentioned in the work. The informed consent of the patients was obtained for the research. No conflict of interests was declared by the authors.

Effects of different loading on the bifurcation of annular elastic rods: Theory vs. experiments

The bifurcation problem of a circular Euler–Bernoulli rod subject to a uniform radial force distribution is investigated under three distinct loading conditions: (i.) hydrostatic pressure, (ii.) centrally-directed, and (iii.) dead load. Previous studies on this apparently ‘familiar’ structural problem have yielded controversial results, necessitating a comprehensive clarification. This study shows that results previously labelled as ‘correct’ or ‘wrong’ simply refer to different external constraints, whose presence becomes necessary only for the two latter loads, (ii.) and (iii.). Moreover, the paper presents the first experimental realization of a circular rod subjected to centrally-directed loads. The experimental findings align with the theoretical predictions and show the exploitation of a new type of load acting on a continuous structural element. The feasibility of this load is demonstrated through the use of inextensible cables and opens the way to applications in flexible robotics when cables are used for actuation.

Shape-retaining beam-like morphing structures via localized snap through

In this study, we present a concept of morphing structure – featuring an arch mounted on a compliant base – that can be reconfigured via snap-through buckling and leverages bistability to retain its morphed shape. We show that one-dimensional arrays of such units yield beam-like structures that, upon localized snapping, can attain multiple, morphologically distinct stable shapes. Units are modeled using discrete elastic rods, a reduced-order formulation for beams and beam structures, and the results are validated via experiments. We leverage our model to understand the influence of the geometrical design parameters on the response and final shape of a unit. We then show that the morphed shapes of arrays of units can be predicted by concatenating results of simulations on single units, and leverage this idea to inverse-design structures that can be snapped into target stable shapes. Ultimately, our work suggests an up-scalable way to create shape-retaining morphing structures with target stable shapes.