A graphical criterion for the instability of elastic equilibria under multiple loads: with applications to drill-strings

We present a new instability theorem which yields stability predictions for any multi-degree-of-freedom elastic system subjected to two or more loads. Predictions are based solely on the form of the equilibrium surfaces in the space of loads and their corresponding displacements. We apply this to the torsional buckling of an elastic rod which can be free or contacting a cylindrical casing, as with a drill-string within its bore-hole. We investigate the stability of the post-buckling states of the rod under an axial load and a twisting moment acting through the end rotation. In a 4D space spanned by the loads and their displacements, we consider all loading situations in which two variables are controlled, the remaining two being passive responses. Cases of particular interest are totally dead loading and mixed loading with the control of one load and the other displacement. Stability properties are uncovered using a circle-signature technique. The new graphical criterion is useful when varying wall contact hinders a conventional stability analysis, and in experiments where the behaviour under dead loads can be inferred from tests under rigid loading.