Traveling Waves in Elastic Rods with Arbitrary Curvature and Torsion

The dynamic Kirchhoff equations, describing a thin elastic rod of infinite length, are considered in connection with the study of the conformations of polymeric chains. A novel special traveling wave solution that can be interpreted as a conformational soliton propagating at constant speed is obtained, featuring arbitrary non-constant curvature and torsion of the rod, in the simple case of constant cross-section, homogeneous density and elastic isotropy. This traveling wave corresponds to a specific constraint on the twist-to-bend ratio of the constant stiffness parameters, which in turn appears to be compatible with the experimental evidence for the mechanical properties of real polymeric chains. Due to such a constraint, the square of the velocity of the solitary wave is directly proportional to the bending stiffness and inversely proportional to the density and to the principal momentum of inertia of the rod. Several applications to the study of conformational changes in polymeric chains are given.