Non-Material Finite Element Modelling of the Bending of a Rod, partially inserted in a Flexible Sleeve with Intrinsic Curvature

Abstract

We consider frictionless contact between an elastic rod and a flexible sleeve. The rod is partially inserted into the sleeve, thus building a compound rod with piece-wise constant bending stiffness and moving boundaries between the three segments: both the rod and the sleeve contribute to the bending stiffness in the overlapping segment. The intrinsic (natural) curvatures of the rod and the sleeve act as as load factors, resulting into bending, relative sliding and change of length of the segments. Concentrated contact interactions (configurational forces) at the transition points repel the rod and the sleeve from each other, eventually causing full ejection when the intrinsic curvatures exceed a threshold. To investigate this behavior, we apply a problem specific non-material finite element discretization of each segment of the elastic structure. Static equilibria are sought by augmenting the set of nodal degrees of freedom with the unknown configurational parameter, which determines sliding motion. After demonstrating the mesh convergence of the computational model, we investigate the parameter space and seek domains of existence of equilibria with non-vanishing overlapping segment. The parameter space includes the intrinsic curvatures of the rod and of the sleeve and also the initial length of the overlapping segment.