Serpentine locomotion through elastic energy release

Abstract

A model for serpentine locomotion is derived from a novel perspective based on concepts from configurational mechanics. The motion is realized through the release of the elastic energy of a deformable rod, sliding inside a frictionless channel, which represents a snake moving against lateral restraints. A new formulation is presented, correcting previous results and including situations never analysed so far, as in the cases when the serpent's body lies only partially inside the restraining channel or when the body has a muscle relaxation localized in a small zone. Micromechanical considerations show that propulsion is the result of reactions tangential to the frictionless constraint and acting on the snake's body, a counter-intuitive feature in mechanics. It is also experimentally demonstrated that the propulsive force driving serpentine motion can be directly measured on a designed apparatus in which flexible bars sweep a frictionless channel. Experiments fully confirm the theoretical modelling, so that the presented results open the way to exploration of effects, such as variability in the bending stiffness or channel geometry or friction, on the propulsive force of snake models made up of elastic rods.

Pseudomomentum: origins and consequences

“…An early theoretical investigation is that of Kienzler and G. Herrmann [75], who considered discontinuities in stiffness in a beam. More recently, Bigoni and co-workers [49,76,77] have performed a very interesting series of experiments involving both continuous and discontinuous variation in the bending and torsional stiffness of rods as well as confinement conditions imposed on these rods. This group has also offered a theoretical analysis that, we believe, incorrectly conflates forces and material forces, lumping them both under a general heading of configurational or "Eshelby-like" effects.…”

“…In this section, we present our perspective, which is heavily influenced by the analyses of Kienzler and G. Herrmann [75] and O'Reilly [19,78,79], and partially laid out in prior publications [80,81]. After presenting the bulk and singular balance laws for momentum and pseudomomentum for a non-uniform planar Euler elastica, we apply our approach specifically to the problem of planar serpentine locomotion of a rod through a curved channel [48,49], and attempt to delineate which forces appearing in the problem are actual forces and which are configurational forces. We demonstrate that the propulsive material force on the confined rod can be obtained directly by integrating the pseudomomentum balance, and obtain reaction forces at points of geometric and material discontinuity from the singular pseudomomentum balance without any appeal to micromechanical arguments [49].…”

“…In Sect. 5, we consider the bulk and singular balance laws for quasistatic elastica with inhomogeneous bending stiffness, and apply these to understand the propulsive and reaction forces observed in recent studies of confined rods [48,49]. This is perhaps the best illustration of the potential power of the pseudomomentum balance, which provides a simple, almost effortless, derivation of the propulsive "force" on the body after a reasonable prescription for singular sources is provided by analogy with the bulk balance law.…”

Transverse vibrations analysis of a beam with degrading hysteretic behavior by using Euler-Bernoulli beam model

“…For the vibration of a hysteretic cantilever beam with a concentrated mass on top, in [14] and [15], it was considered an empiric model similar to an SDOF system with time dependent visco-elastic characteristic, to approximate the effect of beam structural degradation in the range of its lowest mode. For some theoretical and experimental results, we refer the reader to [1], [5], [10] and [11].…”

Analytical expression of elastic rods at equilibrium under 3D strong anchoring boundary conditions

“…Elastic rod models are used in different domains at different length-scales : supercoiled DNA modeling [1], animal locomotion [2,3] or vibrissal system [4,5] simulation, chirurgical intervention [6], nonlinear springs designing [7] and marine cables guiding [8] are examples. In some applications as resolving the conformation of single-stranded nucleic acid structures [9][10][11][12], it is required to compute equilibrium configurations of elastic rods under specified geometric constraints.…”