Slender Elastic Rods Research Articles

MULTIPLE HELICAL PERVERSIONS OF FINITE, INTRISTICALLY CURVED RODS

We investigate mechanical spatial equilibria of slender elastic rods with intristic curvature. Our work is, to some extent, motivated by papers [Goriely & Tabor, 1998; Goriely & McMillen 2002]. There such rods of infinite length were recently studied to quantify the behavior of botanical filaments. In particular, an adequate explanation for the existence of helical perversions (the transition between helical segments of opposite handedness) is provided in [Goriely & Tabor, 1998]. However, this theory fails to describe multiple perversions, which can be observed in Nature. In contrast we formulate a two-point boundary-value problem describing rods of finite length with initial curvature and clamped ends. We identify trivial solutions as straight configurations and also k-covered circles, rigorously establish the existence of local bifurcations, and then compute global solutions via the Parallel Hybrid Algorithm [Domokos & Szeberényi, 2004] to find spatially complex equilibria characterized by multiple perversions. Based on computational results and the White–Fuller theorem [White, 1969; Fuller, 1971; Calugareanu, 1961] we describe a heuristic global picture of the bifurcation diagram, which can serve as an explanation for the evolution of physically observable tendril shapes.

Mechanical energy and momentum of wave pulses in a dispersionless, lossless elastic medium, according to the linear theory

We re-examine the linear theory of wave propagation through an elastic string under uniform tension or a slender elastic rod from a perspective that focuses on the flow of mechanical energy and mechanical momentum. Continuity equations are established for the flow of energy and momentum, leading to two boundary conditions for the net wave displacement. The important special case of a small amplitude pulse of arbitrary shape traveling through a uniform slender medium joined to another medium with a different linear mass density is examined in detail. The new boundary conditions lead to the correct relative amplitudes for the reflected and transmitted pulses. We obtain the instantaneous mechanical energy and momentum of the incident, reflected, and transmitted pulses and show that the net mechanical energy and momentum are separate constants of motion. The cases of an incoming pulse described by a Lorentzian and a Gaussian distribution are suggested as problems to be solved by the interested reader.

THERMAL POSTBUCKLING OF SLENDER ELASTIC RODS WITH HINGED ENDS CONSTRAINED BY A LINEAR SPRING

This paper presents the critical buckling temperature and a closed-form analytical solution for the thermal postbuckling behavior of uniformly heated slender elastic rods. The boundary conditions are assumed hinged and nonmovable at one end, whereas at the other end longitudinal displacement is constrained by a linear spring. The thermal strain–temperature relationship is considered nonlinear and the material is assumed to be linearly elastic, homogeneous, and isotropic. Large displacements are included; hence, the formulation is geometrically nonlinear. A set of six first-order nonlinear ordinary differential equations with boundary conditions defined at both ends yields a complex boundary-value problem. The buckling and postbuckling solutions are respectively accomplished by assuming infinitesimal rotations and employing complete elliptical integrals. The results are presented in nondimensional graphs obtained for different values of spring stiffness, slenderness ratios, and nonlinear thermal strain coefficients. It is shown that these three parameters govern the buckling and postbuckling behavior.

Post-buckling analysis of slender elastic rods subjected to terminal forces

This paper presents formulation and solutions for the elastica of slender rods subjected to axial terminal forces and boundary conditions assumed hinged and elastically restrained with a rotational spring. The set of five first-order non-linear ordinary differential equations with boundary conditions specified at both ends constitutes a complex two-point boundary value problem. Solutions for buckling, initial post-buckling (perturbation), large loads (asymptotic) and numerical integration are developed. Results are presented in non-dimensional graphs for a range of rotational spring stiffness, tuning the analysis from double-hinged to hinged–built-in rods.

Cell prestress. II. Contribution of microtubules.

The tensegrity model hypothesizes that cytoskeleton-based microtubules (MTs) carry compression as they balance a portion of cell contractile stress. To test this hypothesis, we used traction force microscopy to measure traction at the interface of adhering human airway smooth muscle cells and a flexible polyacrylamide gel substrate. The prediction is that if MTs balance a portion of contractile stress, then, upon their disruption, the portion of stress balanced by MTs would shift to the substrate, thereby causing an increase in traction. Measurements were done first in maximally activated cells (10 microM histamine) and then again after MTs had been disrupted (1 microM colchicine). We found that after disruption of MTs, traction increased on average by approximately 13%. Because in activated cells colchicine induced neither an increase in intracellular Ca(2+) nor an increase in myosin light chain phosphorylation as shown previously, we concluded that the observed increase in traction was a result of load shift from MTs to the substrate. In addition, energy stored in the flexible substrate was calculated as work done by traction on the deformation of the substrate. This result was then utilized in an energetic analysis. We assumed that cytoskeleton-based MTs are slender elastic rods supported laterally by intermediate filaments and that MTs buckle as the cell contracts. Using the post-buckling equilibrium theory of Euler struts, we found that energy stored during buckling of MTs was quantitatively consistent with the measured increase in substrate energy after disruption of MTs. This is further evidence supporting the idea that MTs are intracellular compression-bearing elements.

On the deflected configuration of a slender elastic rod subject to parallel terminal forces and moments

This paper investigates the three-dimensional configurations of a slender elastic rod of uniform circular cross-section subject to parallel terminal forces and moments. The nonlinear, equilibrium equations for the rod are established for a Cartesian coordinate system and solved analytically without linearization. Consequently, the results are applicable for large nonlinear elastic deformations.

Acceleration waves in an extensible elastica

We consider here the propagation of acceleration waves in planar elastic rods that are capable of bending and stretching. The moment-curvature and force-stretch relations used have sufficient generality as to afford a description of materially inhomogeneous, geometrically non-uniform, slender elastic rods undergoing finite deformations. It is shown that either the curvature of the rod or its rate of stretch suffers a jump discontinuity at an acceleration wave. The former wave is shown to have the characteristics of a longitudinal shock wave propagating into an inhomogeneous elastic body. Speeds of propagation and conditions governing the evolutionary behavior of these waves are derived. Discontinuities induced by these waves, and higher order waves are also considered.

On waves in slender elastic rods

On waves in slender elastic rods

On Stability Equations of Spatial Rods - A Technical Theory

General nonlinear stability equations of naturally curved and pretwisted slender elastic rods are derived by means of a technically heuristic approach. The derived equations take account of such effects as deformation prior to loss of stability, shearing deformation and the influence of loading behaviour. Through systematic simplifications stability equations of planar arches and straight rods are derived and are shown to be in agreement with the corresponding classical equations. Some aspects of loading behaviour and instability modes, as related to conservative and non-conservative nature of the problem, are also studied.

Buckled Elastica at Contact: A Perturbation Analysis

We consider the problem describing the shape assumed by a slender elastic rod clamped at both ends and subjected to an axial force. For the case of the first bending mode the elastica comes into contact with itself at a critical value of the applied pressure $P_c $. We develop a perturbation analysis for the shape function in terms of the expansion parameter $P - P_c $ for $P > P_c $ and $P - P_c $ small relative to $P_c $. A nonlinear change of independent variables is used to freeze the contact point.

Stresses in and around slender elastic rods and platelets of different modulus in an infinite elastic medium under uniform

Approximate solutions for the stresses around slender elastic rods and platelets with moduli different from that of the infinite elastic matrix in which they are built-in have been obtained by the method of Eshelby, for the case of uniform distant axial strain in the matrix parallel to the rod or platelet. The results show that, as expected, the interface tractions are strongly concentrated at the ends of these heterogeneities.

Stability of Nonlinear Oscillations of an Elastic Rod

The stability of plane steady oscillations of a slender elastic rod subjected to lateral harmonic excitation is investigated. For certain values of excitation amplitudes and frequencies the planar response is unstable, and nonplanar motions are parametrically excited. Steady nonplanar motions are characterized by each point on the rod centerline tracing an elliptical path perpendicular to the rod axis. The investigation is restricted to cantilevered rods with nearly equal principal moments of inertia of the cross-section, and excitation is always in the direction of one of the principal axes of inertia. The effects of damping and of slight variations of the natural frequencies associated with motions in the two principal planes are investigated. Supporting experimental evidence is presented. Finally, the principal features of the transversely driven rod response are compared with the response for the same type rod excited axially.

Stability of Nonlinear Oscillations of an Elastic Rod

The stability of plane steady oscillations of a slender elastic rod subjected to lateral harmonic excitation is investigated. For certain values of excitation amplitudes and frequencies, the planar response is unstable, and nonplanar motions are parametrically excited. Steady nonplanar motions are characterized by each point on the rod centerline tracing an elliptical path perpendicular to the rod axis. The investigation is restricted to cantilevered rods with nearly equal principal moments of inertia of the cross section, and excitation is always in the direction of one of the principal axis of inertia. The effects of damping and of slight variations of the natural frequencies associated with motions in the two principal planes are investigated. Supporting experimental evidence is presented. Finally, the principal features of the transversely driven rod response are compared with the response for the same type rod excited axially.

The bending of a thin rod having initially uniform curvature

A relation is established between the forms of the elastica obtained when equal and opposite forces are applied to a slender elastic rod having initially the form of a circular arc. Sets of curves are obtained which demonstrate the manner in which several mathematical forms may be involved in a single instance. The case of a circular hoop severed at one point is treated approximately. The work is also applied to the behaviour in practice of slender columns under compression.