On a compressed spatial elastica constrained inside a tube

Abstract

In this paper, we study the deformation of a clamped-clamped elastica under edge thrust and constrained inside a straight tube. The governing equations are formulated within the framework of director theory and solved by the shooting method. The deformations calculated from the elastica model are compared with those computed from small-deformation theory. For a relatively slender tube, the early stage of the deformation sequence is similar to the one obtained from small-deformation theory. They are one-point, two-point, three-point, and point-line-point contact deformations. However, some fundamental differences exist between these two theories even in this early stage of deformation. According to small-deformation theory, the constrained rod jumps from planar one-point contact to helical two-point contact suddenly. In elastica model, the planar one-point contact evolves to spatial one-point contact first and transforms to two-point contact smoothly. In small-deformation theory, the point-line-point contact is the final stage of deformation. In elastica model, on the other hand, the deformation may transform to other shapes as the edge thrust increases further. Generally speaking, the difference between small-deformation theory and elastica model grows as the radius of the constraining tube becomes larger. In the case when the ratio of the tube radius and the beam length is larger than 0.384, the constraining tube has no effect on the elastica deformation.