Rod Equation Research Articles

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.

Local-in-space wave breaking criteria for a generalized rod equation

A generalized rod equation including the celebrated Camassa–Holm shallow water model is considered. The blow-up features of the equation are discussed on the line R. Using the method to construct the Lyapunov functions, local-in-space wave breaking criteria to the equation are established. Our wave breaking results are only involved in the initial values V0(x0) and V0′(x0) at a single local point x0∈R.

An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod

Abstract The free longitudinal vibrations of a rod are described by a differential equation of the form ( P ( x ) y ′ ) ′ + λ P ( x ) y ( x ) = 0 {(P(x)y\prime)^{\prime}+\lambda P(x)y(x)=0} , where P ⁢ ( x ) {P(x)} is the cross section area at point x and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form 𝐀 ⁢ Y = Λ ⁢ 𝐁 ⁢ Y {\mathbf{A}Y=\Lambda\mathbf{B}Y} , where 𝐀 {\mathbf{A}} and 𝐁 {\mathbf{B}} are Jacobi and diagonal matrices dependent to cross section P ⁢ ( x ) {P(x)} , respectively. Then we estimate the eigenvalues of the rod equation by correcting the eigenvalues of the resulting matrix eigenvalue problem. We give a method based on a correction idea to construct the cross section P ⁢ ( x ) {P(x)} by solving an inverse matrix eigenvalue problem. We give some numerical examples to illustrate the efficiency of the proposed method. The results show that the convergence order of the method is O ⁢ ( h 2 ) {O(h^{2})} .

The stick-slip bending behavior of the multilevel helical structures: A 3D thin rod model with frictional contact

The multilevel helical structures in various engineering and natural fields offer excellent deformation flexibility and load bearing capabilities. Understanding the interplay between the local frictional contact and the geometric characteristics of the helical structure under complex external loads has attracted considerable interest. In this work, the effect of local frictional contact behaviors on the bending in multilevel helical structures is investigated by using a combination of theoretical modeling, finite element (FE) simulations, and experiments. In the case of pure bending, the kinematic parameters of the bent multi-stage helix are derived concisely by the idea of the kinematic analogy. The bending stiffness of the multi-stage helix is further obtained. In the case of the combined tension/torsion and bending, the 3D thin rod model incorporating Coulomb’s friction is established to describe the mechanical responses. It is found that the relationship between equivalent bending stiffness and the laying angle exhibits nonlinearity. A comparison with the classical Papailiou model reveals that, for helical structures at large laying angles, the influence of friction is primarily determined by the internal force in the tangential direction, which is the core assumption of the Papailiou model. However, in the case of small laying angles, the helical twisting characteristics and the contribution of the internal forces and moments in the other two directions (normal and binormal directions) to the friction cannot be ignored. Subsequently, a multilevel frictional contact transmission formulation is proposed according to the force action–reaction principle. Based on the above formulation, the non-simplified thin rod equations with Coulomb’s friction are extended to describe the multilevel stick-slip bending behaviors of the second stage cable (3*3). The dissipation capacity of helical structures is evaluated quantitatively under the hysteretic bending. Finally, the theoretical model is verified by FE simulations and experimental results. This work provides insights for unveiling the intrinsic relationship between the nonlinear bending and local frictional contact behaviors in the multilevel helical structures.

Numerical analysis of nonlinear Volterra integrodifferential equations for viscoelastic rods and plates

Numerical analysis of nonlinear Volterra integrodifferential equations for viscoelastic rods and plates

Local buckling behavior and reinforced mechanisms of short circular timber columns confined with bamboo scrimber

Developing sustainable high-performance bamboo scrimber (BS) as structural and reinforcing materials is significant. BS was used as reinforcing material to increase ancient timber columns' load-bearing capacity and safety via a damage-free strengthening method. The BS panels were fastened to the surface of the wooden columns by spaced steel strips. The spacing of the steel strips was the important factor in balancing the appearance of the confined reinforced timber columns with maximizing the strength of the confined panels. In this study, eight short circular wooden columns reinforced with BS with varying spacing of steel strips were fabricated and tested under axial compression. Each specimen's damage characteristics, load-carrying capacity performance, and strain coordination were analyzed and compared. The reinforcement method has a significant effect, and the reinforcement system of all specimens provides an additional load-bearing capacity of more than 48.3 % with about 38 % increase in cross-section area. The load-bearing capacity decreases significantly with increasing the spacing of steel strips. When the steel strips' distance is more than 363 mm (the corresponding slenderness ratio is 41), the confined panel showed significant buckling behavior from the beginning of loading. In contrast, the confined panels of the other circular reinforced timber column showed buckling and load drop only near the peak load, which takes full advantage of the strength of the BS material. The strain growth trends of the confined panels and core column were similar, although the strain of the confined panels' bending offsets some of the compressive strain, which was evidence that they were synergistically stressed under axial compression. The part of confined slats between the steel strips was intercepted as a compression bar, and the commonly used equation for the load-bearing capacity of a flexible bar was introduced. Various flex rod equations based on material properties were calculated and compared to the test results. The calculations showed that the buckling equation suggested by Euler and CSA086-19 had the best agreement with BS materials with similar proportional limits and ultimate strength. BS as a flex rod was suggested with a slenderness ratio not exceeding 32, and a general equation for calculating the steel strip spacing considering the material thickness was proposed for construction reference.

Towards realistic nonlinear elastic bending behavior for cable simulation

AbstractThis contribution aims to characterize and model the nonlinear elastic behavior of cables for reliable simulations. To simulate the nonlinear elastic behavior of cables, we use an iterative method, where at each step, an algorithmic local bending stiffness constant is used and updated according to the current cable state. We also formulate an inverse problem to determine the properties of real cables. By solving the inverse problem, the nonlinear elastic behavior for given measurement data is identified, yielding a curvature‐dependent bending stiffness characteristic. In addition, we propose an alternative method based on the balance equations for rods in static equilibrium to identify the bending stiffness characteristic. We apply both methods to experimental data, and the results are compared and discussed.

A Group-Theoretic Approach to the Bifurcation Analysis of Spatial Cosserat-Rod Frameworks with Symmetry

We present a general approach to the bifurcation analysis of spatial framework structures with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of geometrically exact frameworks is not straightforward. We consider spatial structures comprising assemblages of Cosserat rods; the main difficulty arises from the nonlinear configuration space, due to the presence of (cross-sectional) rotation fields. We avoid this via a single-rod formulation, developed earlier by one of the authors, whereby the governing equations are embedded in a slightly enlarged linear space. The field equations for a framework then comprise the assembly of all such rod equations, supplemented by compatibility and equilibrium conditions at the joints. We demonstrate their equivariance under the natural symmetry group action, and the implementation of group-theoretic methods is now clear within the enlarged linear space. All potential generic, symmetry-breaking bifurcations are predicted a-priori. We then employ an open-source path-following code, which can detect and compute simple, one-dimensional bifurcations; multiple bifurcation points are beyond its capabilities. For the latter, we construct symmetry-reduced problems implemented by appropriate substructures. Multiple bifurcations are rendered simple, and the path-following code is again applicable. We first analyze a simple tripod framework, providing all details of our methodology. We then treat a more complex hexagonal space frame via the same approach. Both structures exhibit simple and double bifurcation points.

Analysis of Axially Loaded Piezoelectric Semiconductor Rods with Geometric Nonlinearity

Piezoelectric semiconductor (PS) nanostructures have a huge application potential in flexible electronic devices. We study the nonlinear multi-field coupling mechanical behaviors of axially loaded PS rods by taking the von Kármán type of nonlinear strain–displacement relations into account. The one-dimensional equations for extensional PS rods with the geometric nonlinearity are presented. The analytical solutions for an axially loaded PS rod with open-circuit and electrically isolated boundary conditions at the two ends are obtained based on the classical perturbation method. The zeroth-order perturbation solution is exactly the same as the linearized solution. The influences of the first- and second-order solutions on the multi-field coupling responses of the PS rods under different axial loads are investigated.

A uniform framework for the dynamic behavior of linearized anisotropic elastic rods

A uniform framework for the dynamic behavior of a rod composed of a linearized anisotropic material is constructed based on an asymptotic reduction method. Taylor–Young series expansions of the displacement vector and stress tensor are adopted at the beginning. By substituting the series expansions into the three-dimensional (3D) equilibrium equations and the lateral traction condition followed by proper manipulations, four scalar rod equations are obtained for the leading-order displacement vector and the twist angle of the central line. The associated one-dimensional (1D) boundary conditions are obtained through the virtual work principle. One key feature of the present theory is the establishment of recursive relations so that most of the unknowns from the expansion can be eliminated. Also, all the remainders are retained, so the pointwise error estimate of the equations is established. One important advantage of the present theory is that no ad hoc assumption on the forms of displacement and external loading is adopted. Therefore, the rod theory provides a uniform framework to study dynamic behaviors. As an illustrative example, the natural vibration of a linearized isotropic rod with fixed-free boundary conditions is studied. The natural frequency is calculated by applying this uniform framework for the bending, torsional, and longitudinal modes. The obtained results are compared with those of the 3D simulations, Euler–Bernoulli, and Timoshenko beam theories. It turns out that the present theory is accurate enough for moderate and long rods in the bending vibration, and provides explicit results with high accuracy for either long or short rods in torsional and longitudinal vibrations.

A novel reduced model for a linearized anisotropic rod with doubly symmetric a cross-section: I. Theory

A novel reduced model is constructed for a linearized anisotropic rod with doubly symmetric cross-section. The derivation starts from the Taylor expansion of the displacement vector and the stress tensor. The goal is to establish rod equations for the leading order displacement and the twist angle of the mean line of the rod in an asymptotically consistent way. Fifteen vector differential equations are derived from the 3D (three-dimensional) governing system, and elaborate manipulations between these equations (including the Fourier series expansion of the lateral traction condition) lead to four scalar rod equations: two bending equations, one twisting equation, and one stretching equation. Also, recursive relations are established between the higher order coefficients and the lower order ones, which eliminate most of the unknowns. Six boundary conditions at each edge are obtained from the 3D virtual work principle, and 1D (one-dimensional) virtual work principle is also developed. The rod model has three features: it adopts no ad hoc assumptions for the displacement form and the scalings of the external loadings; it incorporates the bending, twisting, and stretching effects in one uniform framework; and it satisfies the 3D governing system in a point-wise manner.

Designing and Calculating the Nonlinear Elastic Characteristic of Longitudinal-Transverse Transducers of an Ultrasonic Medical Instrument Based on the Method of Successive Loadings.

This paper presents a numerical method for studying the stress–strain state and obtaining the nonlinear elastic characteristics of longitudinal–transverse transducers. The authors propose a mathematical model that uses a direct numerical solution of the boundary value problem based on the plain curved rod equations in Matlab. The system’s stress–strain state and nonlinear elastic characteristic are obtained using the method of successive loadings based on the curved rod’s linearized equations. For most ultrasonic instruments, the operating frequency of ultrasonic vibrations is close to 20 kHz. On the other hand, the received own oscillation frequencies are close to the working range. Using the method of successive loadings in the mathematical complex Matlab, a numerical calculation of the stress–strain state of a flat, curved rod at large displacements has been carried out. The proposed model can be considered an initial approximation to the solution of the spatial problem of the longitudinal–torsional transducer.

Blow-up issues for the hyperelastic rod equation

Blow-up issues for the hyperelastic rod equation

A slender body theory for the motion of special Cosserat filaments in Stokes flow

The motion of filament-like structures in fluid media has been a topic of interest since long. In this regard, a well known slender body theory exists, wherein the fluid flow is assumed to be Stokesian while the filament is modeled as a Kirchhoff rod which can bend and twist but remains inextensible and unshearable. In this work, we relax the inextensibility and unshearability constraints on filaments, i.e., the filament is modeled as a special Cosserat rod. Starting with the boundary integral formulation of Stokes flow involving the filament’s surface velocity and fluid traction that acts on the filament surface, the method of matched asymptotic expansion is used to first obtain a leading-order representation of the boundary integral kernels in the filament’s aspect ratio. We then substitute Fourier series expansion (in filament’s circumferential coordinate) of both the filament’s surface velocity and fluid traction in the aforementioned leading-order representation and further linearize it in the rod’s shear strains to reduce the two-dimensional boundary integral over the filament surface into a line integral over the filament’s centerline. Upon further collecting the coefficients of sine and cosine terms, the zeroth-order Fourier mode yields a line integral equation relating the rod’s centerline velocity with the distributed fluid force that acts on the filament. The presence of line integral makes the relation non-local in nature. On the contrary, the first-order Fourier mode yields a simpler local relation between the rod’s angular velocity and the distributed fluid couple. The line integral equation is shown to reduce to the classical slender body theory when shear strains and axial strain are set to zero. The non-dimensional governing equations of the special Cosserat rod are also derived accounting for the distributed fluid force and distributed fluid couple in them which are solved to obtain the filament motion. The presented theory is demonstrated with an example problem of the tumbling of filaments in background shear flow. We show that for relatively shorter filaments where the effect of shear and axial stretch is more dominant, the obtained results deviate from the ones based on the classical slender body theory.

An ADI finite difference method for the two-dimensional Volterra integro-differential equation with weakly singular kernel

ABSTRACT In this paper, we study the two-dimensional Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain. The memory kernel of this equation makes it not easy to construct an efficient numerical scheme and perform theoretical analysis. The numerical method is considered by the finite difference approach for spatial discretization and Crank–Nicolson (CN) alternating direction implicit (ADI) scheme in the time direction. The integral terms are approximated by the fractional convolution quadrature. We prove that the proposed method is unconditional stable and derive error estimates in norm. The literature reported on the ADI finite difference method of this model is extremely sparse. Numerical results support the theoretical analysis.

An Analytical Approach to an Elastic Circular Rod Equation

The size-dependent longitudinal and torsional dynamic problems for small-scaled rods have importance in two-phase media. The special case of the elastic rod equation such as magneto-electro circular equation are seen in the literature commonly, but in this work, the generalized form of the nonlinear elastic circular equation, which was not studied in the literature, is considered. The exact solutions are obtained via Mathieu approximation method with a novel proposed ansatz. Obtained solutions are discussed and illustrated in details. We believe that the proposed results will be key part of further analytical and numerical studies for waves in the dispersive medium with reaction.

Analysis of 3D composite preform contour locking parameters based on vibration characteristics

To ensure the smooth insertion of the side rod/fiber bundle during the three-dimensional woven preform contour locking process, the influence of the diameter, speed, and other parameters on the free end vibration displacement response of the side rod and the fiber-carrying needle is analyzed. By calculating the kinetic energy and potential energy of the two separately, the dynamic equations of the side rod and the fiber-carrying needle are established based on Hamilton's principle and discretized. The state equations are solved using the Runge–Kutta method, and the free end vibration displacement responses of the side rod and the fiber-carrying needle for different parameters are obtained. The results show that the equilibrium position and the maximum amplitude of the free end of the side rod and the fiber-carrying needle decrease with the increase of the section diameter and the decrease of the axial speed. And the circular tube wall thickness has less impact on the free end vibration displacement of the fiber-carrying needle. The overall analysis shows that the fiber-carrying needle vibration response is smaller than that of the side rod under the same parameters.

Reduced theory for hard magnetic rods with dipole–dipole interactions

Hard magnetic elastomers are composites of soft elastic foundations and magnetic particles with high coercivity. We formulate a theoretical framework to predict the large deformation of a hard magnetic elastomeric rod. In the previous work, the magnetic Kirchhoff rod equations, which constitute a framework for analyzing instabilities for hard magnetic rods, have been developed and validated experimentally for negligible dipole–dipole interactions. Building on previous studies, we derive the magnetic Kirchhoff rod equations with dipole–dipole interactions. The derived equations are integro-differential equations, representing the force and moment balance along the rod centerline that include long-ranged dipole-magnetic force and torque. On the basis of its discrete numerical simulation, we systematically study the effect of the the dipole–dipole interactions strength on the large deformation of hard magnetic rods. In addition, we find that our theory can predict previous experimental results without any adjustable parameters.

Morphological transformation of arched ribbon driven by torsion

The morphological transformation of an arched ribbon driven by torsion is a scientific problem that is connected with daily life and requires thorough analysis. An arched ribbon can achieve an instantaneous high speed through energy transformation and then return to the original shape of the structure. In this paper, based on the characteristics of the ribbon structure, the dynamic mathematical model of the arched ribbon driven by torsion is established from the Kirchhoff rod equation. The variations of the Euler angle of each point on the center line of the ribbon with the arc coordinate s and the rotation angle of the supports ϕ was examined. The relationship between the internal force distribution of each point in the direction of dˆa and the material, cross-sectional properties, and rotation angle of the supports was obtained. We used ABAQUS, a nonlinear finite element analysis tool, to simulate the morphological transformations of the ribbons, verified our theory with simulation results, and reproduced the experimental results of Sano. Furthermore, we redefined the concept of the “critical flipping point” of Sano. In this paper, the dimensional analysis method was used to fit the simulation data. The following relationship between the critical width w∗, thickness h, and the radius R of the ribbon with different cross sections was obtained: w∗=A⋅Rh/R0.6, where A is 3.19 for rectangular cross sections and 3.06 for elliptical cross sections. By analyzing the simulation data, we determined the variation behavior of the out-of-plane deflection of the center point of the ribbon with the radius R, width w, and thickness h. Our research has guiding significance for understanding and designing arched ribbons driven by torsion, and the results can be applied to problems of different scales.

On a consistent rod theory for a linearized anisotropic elastic material II. Verification and parametric study

We have derived a rod theory by an asymptotic reduction method for a straight and circular rod composed of linearized anisotropic material in part I of this series. In the current work, we first verify the derived rod theory through five benchmark Saint-Venant’s problems. Then, under a specific loading condition (line force at the lateral surface with two clamped ends), we apply the rod theory to conduct a parametric study of the effects of elastic moduli on the deformation of a rod composed of four types of anisotropic materials including cubic, transversely isotropic, orthotropic, and monoclinic materials. Analytical solutions for the displacement, axial twist angle, stress, and principal stress have been obtained and a systematic investigation of the effects of elastic moduli on these quantities is conducted, which is the main feature of this paper. It is found that these elastic moduli arise in a certain form and in a certain order in the solutions, which gives information about how to appropriately choose moduli to adjust the deformation. Among the four anisotropic materials, it turns out that the monoclinic material presents the most remarkable mechanical behavior owing to the existence of a coupling coefficient: it yields coupled leading-order rod equations, non-trivial axial twist angle, non-negligible transverse shear deformation, and a more adjustable principal stress along the axis.