The frictionless flexible sliding sleeve

Abstract

The planar mechanics of an elastic rod constrained by a frictionless, flexible sliding sleeve is analyzed. A variational approach is first applied to the equilibrium of an equivalent compound rod system with variable length, leading to a nonlinear boundary value problem. The equilibrium equations determine the deformation kinematics and, through a frictionless sliding condition governed by the Hamiltonian invariant, specify the overlapping length, but they do not reveal interaction forces between the flexible rod and the flexible sliding sleeve. To capture the interaction in detail, the system is modeled as two sub-rods, and both variational and micromechanical methods are employed independently, yielding identical closed-form expressions for the internal forces and moments within the overlapping region. This analysis reveals the presence of tangential concentrated interaction forces of repulsive nature at both ends of the overlapping region. The investigation is complemented by the numerical solution of four case studies, illustrating the broad mechanical behavior of the flexible, frictionless sleeve system. It is found that two different mechanisms may define the maximum bearing capacity, associated with the reciprocal ejection of the two sub-rods: either (i) a quasi-static disappearance of the overlap or (ii) a snap-through instability. The study also shows the possible vanishing of the distributed interaction force, even in non-symmetric configurations. This research establishes a novel theoretical framework for the mechanics of deployable systems and offers insights to advance the design and analysis of structures in fields such as aerospace, robotics, and civil and mechanical engineering.