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Nonlinear stability criteria for tree-like structures composed of branched elastic rods

New necessary conditions for the nonlinear stability of branched tree-like structures composed of elastic rods are presented. The conditions, which are found by examining the second variation of an energy functional, are established by extending Legendre's classical work on this topic. For the branched tree-like structures of interest in this paper, one of the new necessary conditions is the existence of a bounded solution to a set of Riccati equations, which are coupled at branching points. In addition, a set of new necessary conditions are established in the event that the abscissa of the branching point is not fixed. The results are illustrated using a range of examples featuring planar configurations of branched elastic rods where each of the individual rods is modelled using Euler's theory of the elastica.