Refined Equations of Motion of Multilayered Shells with Transversely Soft Fillers under a Medium Bending

As a development of the earlier results, we constructed a refined two-dimensional mathematical model of the dynamic deformation of multilayered plates and shells with transversely soft fillers, based on the classic Kirchhoff–Love model for supporting layers and the hypothesis on similarity of the laws of variation of movements along the thickness of fillers under both static and dynamic loading process. On the ground of this hypothesis, for a transversely soft filler, we derived simplified quasi-static equations of the theory of elasticity, which allow transverse integrating. When integrating the equations to describe the stress-strain state (SSS), we introduced (as in the static problems) two-dimensional unknown functions representing transverse tangent stresses, constant in thickness. Based on the generalized variational Ostrogradskii–Hamilton principle for describing the dynamic processes of deformation with high variability of SSS parameters, we obtained two-dimensional motion equations of general form, where the inertial components have the same degree of accuracy in comparison with the other ones. We simplified the obtained equations for the case of low variability of SSS parameters and considered the problem of free oscillations of a small rectangular multilayered plate, which are characterized by a zero variability of the functions in the tangential directions and are realized in the plate without deformation of the supporting layers.