Torsional waves of finite amplitude in elastic rod

We propose mathematical models generalizing the Coulomb and Vlasov equations of torsional vibrations of rods by taking the geometric nonlinearity into account. In the general case, the nonlinearity is taken into account both in the system of displacements (because the displacement vector in the case of rod torsion can be finite even for small strains) and in the relations between displacements and strains. We analyze nonlinear torsional stationary waves and find the effect of splitting of soliton-like unipolar waves in countercollisions. We also show that, in several cases, the existence of nonlinearities can also induce dispersion and that nonlinear stationary waves can also exist in the absence of dispersion in the linear medium.