On $\Lambda$-Elastica

Abstract

In this paper, we investigate a transition from an elastica to a piece-wised elastica whose connected point defines the hinge angle $\phi_0$ and we call the piece-wised elastica $\Lambda_{\phi_0}$-elastica or $\Lambda$-elastica. Such transition appears in the bending beam experiment when an elastic beam is gradually compressed and at some moment suddenly due to 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.