Non-material FE model for the flexible and extensible sliding sleeve problem

A novel finite element model is proposed for the static problem of a rod that slides without friction in a sleeve, with both components considered flexible and extensible. The tensional compliance gives access to internal axial forces and allows for a direct computation of the tangential contact forces that result from the variability of the contact length. Two configurational parameters are introduced to characterize the horizontal position and extent of the insertion. Upon each of the system’s segments, a Eulerian parametrization is used, followed by an individual normalization. Based on this kinematic description, a finite element discretization is applied featuring elements that neither stick with the material nor remain at their initial spatial position. The discretized model is employed to simulate the system’s response under external loading conditions. Being directly accessible due to the extensibility, the Hamiltonian is numerically shown to retain a constant value for each component even across domains with and without contact. Following an analytic approach, a formula for the tangential contact force is proposed, which also takes precurvature into account. Equipped with the new finite element model, simulations are run that strongly support the suggested relation.