Thin Elastic Rod Research Articles

Growth induced buckling of morphoelastic rod in viscous medium**Project supported by the National Natural Science Foundation of China (Grant Nos. 11772141 and 11262019) and the State Scholarship Fund of China Scholarship Council (Grant No. 201708370030).

Biological growth is a common phenomenon in nature, and some organisms such as DNA molecules and bacterial filaments grow in viscous media. The growth induced instability of morphoelastic rod in a viscous medium is studied in this paper. Based on the Kirchhoff kinetic analogy method, the mechanical model for growing elastic thin rod in the viscous medium is established. A perturbation analysis is used to analyze the stability of the growing elastic rod in the viscous medium. We apply the results into planar growing ring and get its criterion of instability. Take the criterion into DNA ring to discuss the influence of viscous resistance on its instability.

Asymptotic behavior of eigenfrequencies of a thin elastic rod with non-uniform cross-section

We study the eigenvalue problem of the elliptic operator which arises in the linearized model of the periodic oscillations of a homogeneous and isotropic elastic body. The square of the frequency agrees to the eigenvalue. Particularly, we deal with a thin rod with non-uniform connected cross-section in several cases of boundary conditions. We see that there appear many small eigenvalues which accumulate to 0 as the thinness parameter $\varepsilon$ tends to 0. These eigenvalues correspond to the bending mode of vibrations of the thin body. We investigate the asymptotic behavior of these eigenvalues and obtain a characterization formula of the limit equation for $\varepsilon \rightarrow 0$.

Asymptotic analysis of a bit brace shaped junction of thin rods

AbstractWe present a 1‐D model of a junction of five thin elastic rods forming the shape of a bit brace (hand drill), or, a crankshaft. The distinguishing feature of this junction is the existence of the so‐called movable elements, which are rods and knots requiring modifications of the classical asymptotic ansätze. These consist of constant longitudinal displacements on the edges of the skeleton of the junction and affect the transmission conditions at its nodes. We provide asymptotic formulas for the displacements, stresses and elastic energy, as well as error estimates. An exact solution of the model is given for a particular loading.

Dynamic Analysis of Umbilical Cables for Maglev Vibration Isolation Systems

Umbilical cables are the only paths that transmit vibration from the base to the upper platform in a maglev vibration isolation platform. The geometry properties and the vibration transmission characteristics of umbilical cables are crucial for the isolation performance of maglev vibration isolation platforms. The dynamic model of umbilical cables is studied comprehensively based on the nonlinear mechanics of a thin elastic rod, and the model is solved according to a differential quadrature method. A case about a cantilever cable is solved to verify the accuracy, efficiency, and convergence of the solution. The method to obtain an equivalent dynamic model of umbilical cables is analyzed, and the relationship between the excitation parameters and the disturbing forces/moments is obtained, which can be applied in the overall dynamic analysis of maglev vibration isolation systems. Experiments are carried out to verify the dynamic model, the numerical solution, and the equivalent dynamic model. To understand the applicability of the equivalent dynamic model, disturbing forces and moments caused by different frequencies, different amplitudes, and different initial configurations are compared; and the transmission characteristics of umbilical cables are obtained. The research has an important significance on the dynamic analysis of umbilical cables for maglev vibration isolation systems, and the method in establishing an equivalent dynamic model of umbilical cables is applicable.

Asymptotics of Natural Oscillations of Elastic Junctions with Readily Movable Elements

A one-dimensional model of harmonic oscillations of a junction of several thin elastic rods has been developed. In contrast to the classical model of a single rod, the constructed model of the junction is not purely differential but includes new algebraic unknowns and algebraic equations evoked by the so-called readily movable elements of the structure. The asymptotic representations have been found for frequencies and natural modes of the elastic body oscillations and estimates of asymptotic residues have been obtained.

The establishment of a mechanics model of multi-strand wire rope subjected to bending load with finite element simulation and experimental verification

In view of lack of a general mechanics model and theoretical calculation formulas for the bending state of a wire rope at present, a new mechanics modeling method for wire rope under bending is put forward. Firstly, the change law of the secondary helix angle is analyzed, and the reasonable assumptions for the mechanics model are put forward. According to the elastic thin rod theory of Love and the conservation of energy principle, the mechanics model of multi-strand wire rope in bending state is established, and the theoretical formulas of bending stiffness and equivalent elastic modulus are derived. Taking 6 × 7 and 6 × 19 wire ropes as examples, theoretical formulas are applied for calculation, meanwhile, the finite element simulation is carried out using ABAQUS and a special deflection measuring device is designed and manufactured to measure the large deformation deflection of the wire ropes. The theoretical calculation results are compared with the finite element simulation results and the measurement results. The contrast results show that the theoretical results are closer to the measurement results than the finite element simulation results. The maximum deformation error of the wire ropes free end between the theoretical results and the measurement results is 11.16%, which verifies the correctness of the mechanics model.

Numerical simulation and analysis of the dynamic finite element model of the fiber motion in the air spinning process

Under the spinning technology of the air-assisted twisting, this paper established the fiber finite element model, which has the structural and dynamic characteristics of the fiber flexible body, based on the space elastic thin rod unit. In this research, we deduced the static equilibrium equation of the fiber finite element model. The nonlinear geometric displacement large deformation problem of the space elastic thin rod unit was treated by the method of stepwise loading successive approximation, and the finite element simulation and analysis of the fiber flexible body’s movement and deformation during the air-jet twisting were carried out and verified by experiment in this paper. This research presents an effective and feasible theoretical model and method for the flexible fiber twisting formation process and mechanism under the high velocity vortex airflow.

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.

An elastic model of DNA under thermal induced stress

DNAs (deoxyribonucleic acids) have the ability to alter its conformation in response to temperature changes to relieve the internal stress. The mistakenly structured DNA can lead to diseases or deformities. The conformation also influences the binding between DNA and other molecular complexes. In the present paper, we investigate the DNA elasticity under the influence of thermal induced stress by employing Kirchhoff's model of a thin elastic rod. The problem is solved by perturbation method to find equilibrium configurations of DNA at different modes. In addition, the model is validated with existing literature in which DNA is stretched or compressed. The behaviors of the helical structure under various temperatures are investigated with the melting temperature of DNA around 80 °C. This elasticity study of DNA could be a groundwork leading to better understandings on the effects of thermal induced stress to DNA's deformation and relevant biological processes in living cells.

Physical deformation configuration of a spatial clamped cable based on Kirchhoff rods

PurposeIn this study, a modeling method for a clamped deformable cable simulation based on Kirchhoff theory is proposed. This methodology can be used to describe the physical deformation configuration of any constrained flexible cable in a computer-aided design/manufacturing system. The modeling method, solution algorithm, simulation and experimental results are presented to prove the feasibility of the proposed methodology. The paper aims to discuss these issues.Design/methodology/approachFirst, Kirchhoff equations for deformable cables are proposed based on the nonlinear mechanics of thin elastic rods, and the general solution of the equations described by the Euler angles is given in the arc coordinate system. The parametric form solution of the Kirchhoff equations, which is easy to use, is then obtained in a cylindrical coordinate form based on Saint Venant’s theory. Finally, mathematical expressions that reflect the clamped cable configuration are given, and the deformable process is simulated based on an open source geometry kernel and is then tested by a 3D laser scanning technology.FindingsThe method presented in this paper can be adapted to any boundary condition for constrained cables as long as the external force and torque are known. The experimental results indicate that both the model and algorithm are efficient and accurate.Research limitations/implicationsA more comprehensive study must be executed for the physical simulation of more complicated constrained cables, such as the helical spring and asymmetric constraint. The influence of the material properties of the cable on the calculation efficiency must be considered in future analysis.Originality/valueThe semi-analytical algorithm of the cable simulation in cylindrical coordinates is a novel topic and is more accurate and efficient than the common numerical solution.

On global and local minimizers of prestrained thin elastic rods

We study the stable configurations of a thin three-dimensional weakly prestrained rod subject to a terminal load as the thickness of the section vanishes. By $$\Gamma $$ -convergence we derive a one-dimensional limit theory and show that isolated local minimizers of the limit model can be approached by local minimizers of the three-dimensional model. In the case of isotropic materials and for two-layers prestrained three-dimensional models the limit energy further simplifies to that of a Kirchhoff rod-model of an intrinsically curved beam. In this case we study the limit theory and investigate global and/or local stability of straight and helical configurations.

Buckling shape transition of an embedded thin elastic rod after failure of surrounding elastic medium

When the compressive load to a thin elastic rod embedded in an elastic medium exceeds a threshold, the thin rod buckles into an exponentially decaying short wavelength profile to minimize the total energy of the system. As the compressive load continues to increase, the buckling amplitude increases correspondingly, until the rod/medium interface fractures and the short wavelength buckling profile morphs into a different shape as fracture propagates into the surrounding medium. In this study, such shape transition in the presence of surrounding medium failure is investigated using a combined experimental and theoretical approach. We identify the ansatz that can be used to describe the post-fracture buckling profile, and then develop a forward scheme using the energy principle to predict the buckling profile of the thin rod when fracture happens in the medium. We also develop a backward scheme where we use the post-fracture buckling profile to estimate the buckling profile before fracture of the surrounding medium. Comparison of experimental and theoretical results indicates that the modeling framework can be used to characterize the buckling profile transition of a thin elastic rod embedded in a fractured elastic medium.

Nonlinear dynamics behavior analysis of the spatial configuration of a tendril-bearing plant

Tendril-bearing plants appear to have a spiraling shape when tendrils climb along a support during growth. The growth characteristics of a tendril-bearer can be simplified to a model of a thin elastic rod with a cylindrical constraint. In this paper, the connection between some typical configuration characteristics of tendrils and complex nonlinear dynamic behavior are qualitatively analyzed. The space configuration problem of tendrils can be explained through the study of the nonlinear dynamic behavior of the thin elastic rod system equation. In this study, the complex non-Z2 symmetric critical orbits in the system equation under critical parameters were presented. A new function transformation method that can effectively maintain the critical orbit properties was proposed, and a new nonlinear differential equations system containing complex nonlinear terms can been obtained to describe the cross section position and direction of a rod during climbing. Numerical simulation revealed that the new system can describe the configuration of a rod with reasonable accuracy. To adequately explain the growing regulation of the rod shape, the critical orbit and configuration of rod are connected in a direct way. The high precision analytical expressions of these complex non-Z2 symmetric critical orbits are obtained by introducing a suitable analytical method, and then these expressions are used to draw the corresponding three-dimensional configuration figures of an elastic thin rod. Combined with actual tendrils on a live plant, the space configuration of the winding knots of tendril is explained by the concept of heteroclinic orbit from the perspective of nonlinear dynamics, and correctness of the theoretical analysis was verified. This theoretical analysis method could also be effectively applied to other similar slender structures.

Conformal Invariance and Conserved Quantities for Lagrange Equation of Thin Elastic Rod

Conformal Invariance and Conserved Quantities for Lagrange Equation of Thin Elastic Rod

Mechanics model and its equation of wire rope based on elastic thin rod theory

In view of the lack of general mechanics model and theoretical formula for force-deformation calculation of wire rope, a novel modeling method for wire rope mechanics model is presented. The mathematical model of wire rope's secondary helix is established using the Frenet frame and differential geometry. On the basis of that geometrical parameters of secondary helical line is derived from the necessary assumptions and geometrical relationship, using elastic thin rod theory of Love to establish an equivalent mechanics model of wire rope, and the equivalent elastic modulus and equivalent shear modulus calculation formulas are deduced. The deformation and elongation of wire rope under known tension can be calculated by the formulas. The force-deformation calculation of actual wire rope is carried out, and the results show that the maximum deviation of elongation is less than 8% compared with the finite element simulation and experimental measurement. Therefore, the correctness and generality of the model are verified, and it has theoretical and practical significance for the application and analysis of wire rope.

Asymptotics of natural oscillations of a spatial junction of thin elastic rods

A one-dimensional model of an elastic junction that contains hard- and readily-movable thin rods is derived, and asymptotic formulas for eigenvalues with rigorous estimates for remainders are given. In addition to vector functions satisfying classical ordinary differential equations, the model involves algebraic unknowns and algebraic relations corresponding to the longitudinal rigid motion of readily-movable rods. Nous obtenons un modèle 1D d'une jonction élastique qui contient des barres dures fines facilement déplaçables, et nous donnons un développement asymptotique des valeurs propres justifié par des estimations d'erreur rigoureuses. En plus de fonctions vectorielles satisfaisant des équations différentielles ordinaires, le modèle met en jeu des inconnues et des relations algébriques correspondant au mouvement rigide longitudinal des barres.

Stability of anisotropic, naturally straight, helical elastic thin rods

There has been a differential-geometric interpretation which associates each elastic thin rod with a curve on the three-dimensional unit sphere equipped with a Riemannian metric related to the bending and twisting stiffnesses of the rod. In this paper, we exploit this interpretation to study the stability of anisotropic, naturally straight, helical equilibrium rods with clamped ends. Such a rod is called geodesic here if its associated curve is a geodesic, or equivalently, its twisting stiffness equals one of the bending stiffnesses. We establish criteria for geodesic equilibria to be stable and possibly unstable separately, and develop a scheme predicting unstable non-geodesic equilibria. We also present an example to emphasize the necessity of examining whether a helical equilibrium rod is geodesic or not when one is concerned with its stability.

Методика расчета механики систем связанных тонких упругих стержней по дифференциальной модели

Методика расчета механики систем связанных тонких упругих стержней по дифференциальной модели

Parametric resonances in the problem of longitudinal impact on a thin rod

The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.

Analysis of DNA equilibrium configuration under interfacial traction

Kirchhoff thin elastic rod models are important in the study of the mechanism determining the configurations of flexible structures not only at the macroscopic but also at the microscopic scale. In this study, the energy balance method has been well applied to analyze the configuration of a DNA elastic rod in the presence of interfacial traction. An approximate solution for the shape equations has been obtained, and the relationship between the interfacial factor and the configuration of the DNA segment is derived. The results may provide an explanation for the onset of the formation of kinks in DNA when immersed in a solution.