A new approach for studying the transient response of thin-walled beams of open profile with Cosserat-type micro-structure

In the present paper, the dynamic response of a three-layered thin-walled spatially curved beam of open profile is investigated. As a result of the homogenization of heterogeneous rheological parameters of such a beam, its behaviour could be described by the dynamic constitutive Cosserat equations characterizing a general isotropic, linear micropolar material. Under the action of nonstationary excitations on the beam, plane transient waves (surfaces of discontinuity) are generated in the beam. By the help of the generalized Hadamard-Thomas conditions of compatibility, the velocities of four types of transient waves are found from the set of dynamic Cosserat equations, and the recurrent equations have been obtained which allow one to determine the discontinuities in the desired values and discontinuities in the arbitrary order time-derivatives of these values on each of four waves. This procedure enables one to construct the desired values in terms of the ray series behind the wave fronts up to the boundary where the waves have been generated. The superposition of the ray series for four types of waves allows one to define completely each of the values to be found within the entire disturbed domain. As an example illustrating the proposed procedure for solving boundary-value dynamics problems for beams subjected to transient loading, the problem of the normal impact of a long elastic rod with a rounded end upon a lateral surface of a three-layered thin-walled beam of open profile is considered.