Elliptic integral solutions of spatial elastica of a thin straight rod bent under concentrated terminal forces

Abstract

In this article we solve in closed form a system of nonlinear differential equations modelling the elastica in space of a thin, flexible, straight rod, loaded by a constant thrust at its free end. Common linearizations of strength of materials are of course not applicable any way, because we analyze great deformations, even if not so large to go off the linear elasticity range. By passing to cylindrical coordinates ρ, θ, z, we earn a more tractable differential system evaluating ρ as elliptic function of polar anomaly θ and also providing z through elliptic integrals of I and III kind. Deformed rod's centerline is then completely described under both tensile or compressive load. Finally, the planar case comes out as a degeneracy, where the Bernoulli lemniscatic integral appears.

Flexible and Transparent Gas Molecule Sensor Integrated with Sensing and Heating Graphene Layers

“…Although there is slight variation in the decay curves as the bending strain increased from 0 to 1.4%, the sensing curvature is not distorted. Note that the bending strain ( ε ) was extracted from the bending geometry equation as ε = ( δ p + δ g )/2 R c × (1 + 2 η + χη 2 )/(1 + χη + χη + χη 2 ), where η = δ g / δ p , χ = Y g / Y p , and R c is bending radius, and δ p , δ g , and Y p , Y g are thickness and Young's modulus for PES substrate and graphene, respectively . In Figure c, we display ΔR/R 0 ratio as a function of strain at three different conditions corresponding the (1) off, (2) on, and (3) reset in Figure b.…”

Maximum attainable energy in a fixed-frequency cyclotron

“…Equation (13) shows that at the beginning, when n = 0 and E = E 0 , we have q = 2eV . Hence ( 13) can be integrated with the result…”

Explicit Parameterization of Euler’s Elastica

“…The elastic behaviour of roads and beams which attracts a continuous attention since the time of Galileo, Bernoulli and Euler has generated recently a renewed interest in plane [2,3,15], space [18] and space forms [1,11]. The first elastic problem was posed by Galileo around 1638 who asked the question about the force required to break a beam set into a wall.…”