Snap boundary of self-contacted planar elastica under prescribed end rotations

Abstract

The deformation of an initially straight elastic strip can be determined by prescribing the two end angles and the distance between the two ends. At certain end angles and end distance, snapping motion of the elastica may occur. A collection of these end angle pairs is called the snap boundary of the planar elastica. In the snap boundary theory developed previously (Cazzolli and Corso, 2019) self-intersection of the elastica was admitted. In this paper we extend the theory in Cazzolli and Corso (2019) by excluding self-intersection and admitting only self-contact. We define a dimensionless end distance as the ratio between the physical end distance and the total length of the planar elastica. When the dimensionless end distance is great than 0.246, the planar elastica does not involve self-intersection or self-contact. Therefore, the snap boundary theory previously developed in Cazzolli and Corso (2019) still holds. On the other hand, when the dimensionless end distance is smaller than 0.246 the snap boundaries for self-intersection and self-contact will be different. In this range one of the two stable configurations inside the snap boundary may contact itself. When the dimensionless end distance is further reduced to smaller than 0.151, both of the two stable equilibrium configurations may involve self-contact.