Buckling between soft walls: Sequential stabilisation
through contact
Z.K. Wang & G.H.M. van der Heijden
Motivated by applications of soft contact problems such as guidewires used
in medical and engineering applications, we consider a compressed rod
deforming between two parallel elastic walls. Free elastica buckling modes
other than the first are known to be unstable. We find the soft constraining
walls to have the effect of sequentially stabilising higher modes in multiple
contact by a series of bifurcations in each of which the degree of
instability (the index) is decreased by one. Further symmetry-breaking
bifurcations in the stabilisation process generate solutions with different
contact patterns that allow for a classification in terms of binary symbol
sequences. In the hard-contact limit all these bifurcations collapse into
highly-degenerate `contact bifurcations'. For any given wall separation at
most a finite number of modes can be stabilised and eventually, under large
enough compression, the rod jumps into the inverted straight state. We chart
the sequence of events, under increasing compression, leading from the
initial straight state in compression to the final straight state in tension,
in effect the process of pushing a rod through a cavity. Our results also
give new insight into universal features of symmetry-breaking in higher-mode
elastic deformations.
We present this study also as a showcase for a practical approach to
stability analysis based on numerical bifurcation theory and without the
intimidating mathematical technicalities often accompanying stability
analysis in the literature. The method delivers the stability index and can
be straightforwardly applied to other elastic stability problems.
keywords: buckling, constrained beam, Winkler foundation, large deformation,
bifurcation, symmetry-breaking, stability
Proc. R. Soc. A 477, 20210106 (2021)