The post-buckling behavior of a beam constrained by nonlinear springy walls

The post-buckling behavior of a beam that is subjected to lateral constraints is of relevance to a range of medical and engineering applications, such as endoscopic examination of internal organs, the insertion of a guidewire into an artery in stent procedures, root growth, deep drilling, and more. In this paper we address a disconnect between the existing literature and the reality of these systems, in which the lateral constraints are flexible and experience nonlinear deformations. As a step towards bridging this gap, we consider a beam undergoing planar deformations that is laterally constrained by a non-linear springy wall, i.e. a wall that is laterally pushed by the beam against a non-linear spring. Based on a simplified mathematical model, we obtain closed form analytical solutions, which provide valuable insights and intuition. For example, we show that important features of the behavior, such as transition from point contact to line contact and switching to the next mode, are dictated solely by a non-dimensional force, regardless of all other parameters of the system, and that the full description of the behavior is possible by means of two non-dimensional quantities that describe the relative stiffness of the nonlinear spring compared to that of the beam. The results also highlight the fundamental differences between the behavior with a stiffening spring or with a softening spring, such as the number of attainable modes and the monotonicity of the overall force-displacement relation. These results are then validated by experiments.