On $\Lambda$-Elastica

TL;DR

This paper develops a mathematical theory describing the transition from classical elastica to piece-wise elastica with rupture, using elliptic functions, and connects it to experimental observations of elastic beam failure.

Contribution

It introduces a novel mathematical framework for $ ext{Lambda}$-elastica, capturing rupture phenomena in elastic beams with explicit elliptic function representations.

Findings

Explicit shape representations of $ ext{Lambda}$-elastica using elliptic $ ext{zeta}$-functions

Theoretical explanation of shape transition during beam rupture

Numerical validation of $ ext{Lambda}$-elastica shapes

Abstract

In this paper, we investigate a transition from an elastica to a piece-wised elastica whose connected point defines the hinge angle $\phi_0$; we refer the piece-wised elastica $\Lambda_{\phi_0}$-elastica or $\Lambda$-elastica. The transition appears in the bending beam experiment; we compress elastic beams gradually and then suddenly due the rupture, the shapes of $\Lambda$-elastica appear. We construct a mathematical theory to describe the phenomena and represent the $\Lambda$-elastica in terms of the elliptic $\zeta$-function completely. Using the mathematical theory, we discuss the experimental results from an energetic viewpoint and numerically show the explicit shape of $\Lambda$-elastica. It means that this paper provides a novel investigation on elastica theory with rupture.