On the inextensibility assumption in the stability of elastic rings: overhaul of a traditional paradigm

One of the oldest and most common structural engineering issues is the elastic buckling of circular rings under external pressure, which has a fundamental importance in a number of applications in general mechanics, engineering and bio-physics, just to name a few. Levy is considered to have provided the first significant solution to this problem in 1884, and most stability textbooks make reference to this original solution, which is based on the Euler–Bernoulli beam model. Following this incipit, over the past 140 years a huge number of papers have continued to analyze many special cases and extensions. However, the majority of these studies tend to build on the a priori assumption of inextensibility of the ring centerline without investigating the real significance and extent of this condition. Here, in the framework of a general nonlinear kinematic, the problem is re-examined from its roots, and it is shown that not only the inextensibility paradigm cannot straightforwardly lead to the classic solution in a coherent energy framework but also the extensibility of the ring centerline is necessary to allow a unified and meaningful treatment of buckling and initial postbuckling behavior for a complete variety of cases. On these bases, some facts and results in the literature are rectified and discussed.