The use of the surface location approach in the modeling of periodically nonhomogeneous slender visco-elastic beams

The aim of the study is to develop an analytical reformulation the known theory for thin periodically nonhomogeneous viscoelastic beams, which should to form the basis for the planned research of the full beam dispersion relation. The methodology applied in the paper is similar to that described in paper Tolerance Modelling of Vibrations and Stability for Periodic Slender Visco-Elastic Beams on a Foundation will Damping. Revisiting by Jarosław Jędrysiak, published in scientific journal “Materials” in 2020, but it differs in the way of the use of tolerance modeling in the beam theory. The mathematical tool of considerations is a special choice of the micro-macro decomposition of the beam deflection. It is based on a certain regularization of the displacement field (in the small neighborhood of discontinuity surfaces) and results a certain reformulations of the classical beams theory. Obtained model equations is an alternative proposal for model equations obtained in mentioned paper by Professor Jędrysiak as a certain approximation for periodically nonhomogeneous viscoelastic beams theory as a result of the original tolerance modeling developed. The standardization of the beam deflection field proposed in the presented paper allows us to use the infinite Fourier expansion for deflection field in any region occupied by a homogeneous periodic composite material, which can be modeled in a typical way using the virtual work principle for commonly used beam constrains. The deflection field of the beam is written in the form of an infinite Fourier series, using periodically distributed region of material homogeneity of the beam. Applied method can be viewed as an attempt to use an infinite number of shape functions in the tolerance modeling but at the same time as an equivalent reformulation of the equations of the beam vibration theory. The obtained system of equations is an infinite system of ordinary differential equations for infinitely many Fourier coefficients in the mentioned Fourier expansion of the beam deflection field with respect to the base described in the surface location method.