A generalized computational approach to stability of static equilibria of nonlinearly elastic rods in the presence of constraints

We present a generalized approach to stability of static equilibria of nonlinearly elastic rods, subjected to general loading, boundary conditions and constraints (of both point-wise and integral type), based upon the linearized dynamics stability criterion. Discretization of the governing equations leads to a non-standard (singular) generalized eigenvalue problem. A new efficient sparse-matrix-friendly algorithm is presented to determine its few left-most eigenvalues, which, in turn, yield stability/instability information. For conservative problems, the eigenvalue problem arising from the linearized dynamics stability criterion is also shown to be equivalent to that arising in the determination of constrained local minima of the potential energy. We illustrate the method with several examples. A novel variational formulation for extensible and unshearable rods is also proposed within the context of one of the example problems.