Waves in Constrained Materials and Thin-Walled Beams

The propagation condition for plane infinitesimal waves in constrained elastic materials subjected to arbitrary homogeneous strain is derived and found to be the same as a previously known result for the propagation of acceleration waves. The formal structure of the system of equations leading to the propagation condition is analyzed and extended to include more general situations. In particular, a nonlinear formulation of thin walled beam theory in terms of a constrained multidirected curve is covered by such a generalization. This fact is exploited to derive the frequency equations for small vibrations of thin walled elastic beams of open cross-section about an arbitrary uniformly strained state and to obtain critical loads for Euler and lateral buckling. The results are compared with the classical formulas based on a linearized pre-buckling state and the neglect of shear deformations.