Instability of a penetrating blade

Abstract

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 behaviours: (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 loads 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 behaviours 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.

New phenomena in nonlinear elastic structures: from tensile buckling to configurational forces

“…34, differing only in the boundary conditions at the right end. These structures are similar to those analyzed in Section 2, except for the presence of the Eshelby-like force, arising from the sliding sleeve and which strongly affects the post-critical behaviour and its stability (see Bigoni et al, 2014b).…”

“…A structural system with a free end loaded with an axial dead force was shown to exhibit several surprising and counterintuitive mechanical behaviours by Bigoni et al (2014b) and will not be repeated here. In the following, the governing equations will be obtained for the structures shown in Fig.…”

“…24 This restriction holds only for the calculation of the critical loads. Indeed, equilibrium configurations with p ≥ 1 are possible for non-trivial deformation paths (see Bigoni et al, 2014b). i) since all the coefficients of the cubic equation (192) are real, the following infinite sequence always exists of real roots…”

“…In Bigoni et al (2014b) it is shown that p A cr corresponds to a critical buckling load, while p B cr to a critical restabilization load for which the trivial path returns to be stable. In a similar vein, it can be concluded for the structures now under consideration that: Figure 35.…”

“…Equation (200) 4 represents the equilibrium in the axial direction, revealing the presence of the configurational force B θ eq (l eq ) 2 /2 (Bigoni et al, 2014b). For conciseness, only the first bifurcation mode will be analyzed, so that the symmetry of the problem allows us to consider only the first quarter of the outer blade, similarly to Section 2.2.…”