Instability of a penetrating blade

Application of a dead compressive load at the free end of an elastic rod (the ‘blade’) induces its penetration into a sliding sleeve ending with a linear elastic spring. Bifurcation and stability analysis of this simple elastic system shows a variety of unexpected behaviors: (i) an increase of buckling load at decreasing of elastic stiffness; (ii) a finite number of buckling loads for a system with infinite degrees of freedom (leading to a non-standard Sturm–Liouville problem); (iii) more than one bifurcation load associated to each bifurcation mode; (iv) a restabilization of the straight configuration after the second bifurcation load associated to the first instability mode; (v) the presence of an Eshelby-like (or configurational) force, deeply influencing stability. Only the first of these behaviors was previously known, the second and third ones disprove common beliefs, the fourth highlights a sort of ‘island of instability’, and the last one shows surprising phenomena and effects on stability.