On the modelling of stability problems for thin cylindrical shells with two-directional micro-periodic structure

Abstract

Abstract Thin linearly elastic Kirchhoff-Love-type circular cylindrical shells having a micro-periodic structure in circumferential and axial directions (biperiodic shells) are considered. The aim of this paper is to investigate the effect of a periodicity cell size on the stationary and dynamical stability of such shells. In order to take into account the length-scale effect in some special stability problems, an averaged non-asymptotic biperiodic shell model derived by means of the known tolerance modelling procedure is applied. Governing equations of this averaged model have constant coefficients depending also on a microstructure size, contrary to the starting exact shell equations with periodic, non-continuous and highly oscillating coefficients (the well-known governing equations of Kirchhoff-Love second-order theory of thin elastic shells). The effect of a cell size on critical forces and on frequency equation being a starting point in the analysis of parametric vibrations and dynamical stability is studied. It will be shown that the micro-periodic heterogeneity of the shells leads to the fourth-order ordinary differential frequency equation, which can be treated as a certain generalization of the known Mathieu second-order equation. It reduces to the Mathieu equation provided that the length-scale effect is neglected.