Speaker
Description
The problem of sliding of an elastic rod, partially inserted into a sleeve is attracting more and more researchers’ attention. Dynamics, stability, locomotion are just some of the mechanical phenomena intrinsic to the set-up, whose modelling requires novel analytical and numerical techniques. The behavior of the structure is strongly determined by a configurational force acting between the rod and the sleeve at the transition point separating the overlapping region (rod inside the sleeve) and the free segment. While the investigation of such Eshelby-like forces (named so because of the analogy to fracture mechanics) in the context of structural mechanics dates back to middle 1980s, their role in various problems featuring relative sliding has been understood relatively recently.
In the present contribution, we release the assumption that the sleeve is rigid and consider frictionless contact between an elastic rod and a flexible sleeve, also treated as a rod. Because of the partial insertion, we deal with a compound beam with piecewise constant bending stiffness (as both the rod and the sleeve contribute to the bending stiffness in the overlapping region) and moving boundaries between the three segments. Configurational forces at the transition points repel the rod and the sleeve from each other, eventually causing full ejection when the external loads exceed certain threshold. Considering the outer ends of the rod and of the sleeve to be simply supported and loaded by bending moments, we develop the governing equations for the static equilibrium by means of a variational procedure, in which the length of the overlapping region is an unknown configurational parameter, whose variation is independent. This results into a specific sliding condition, which relates state variables of all three segments and provides a well-posed boundary value problem. Alternatively, stable equilibria can be obtained by means of a non-material finite element formulation. Further investigations on the model demonstrate two scenarios of a catastrophe as the moment load reaches the critical value: the rods may detach either when the overlapping region vanishes, or because of a snapping instability (fold point).
While being powerful in computing equilibria, the compound beam model lacks the ability to predict the forces acting between the rod and the sleeve along the overlapping region and in the transition points. We address this problem in the final part of the talk, demonstrating that it is necessary to include the extensibility of the sub-rods into account to resolve the statical indeterminacy.
