Abstract
The objects of consideration are thin linearly elastic Kirchhoff–Love-type circular cylindrical shells having a periodically micro-heterogeneous structure in circumferential direction (uniperiodic shells). The aim of this contribution is to study certain problems of micro-vibrations and of wave propagation related to micro-fluctuations of displacement field caused by a periodic structure of the shells. These micro-dynamic problems will be analysed in the framework of a certain mathematical averaged model derived by means of the combined modelling procedure. The combined modelling includes both the asymptotic and the tolerance non-asymptotic modelling techniques, which are conjugated with themselves under special conditions. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, governing equations of the combined model have constant coefficients depending also on a cell size. Hence, this model takes into account the effect of a microstructure size on the dynamic behaviour of the shells (the length-scale effect). It will be shown that the micro-periodic heterogeneity of the shells leads to cell-depending micro-vibrations and to exponential waves as well as to dispersion effects, which cannot be analysed in the framework of the asymptotic models commonly used for investigations of vibrations and wave propagation in the periodic structures.
Highlights
Thin linearly elastic Kirchhoff–Love-type circular cylindrical shells with a periodically micro-inhomogeneous structure in circumferential direction are analysed
It will be shown that the periodic micro-heterogeneity of the shells leads to vibrations depending on a cell size and to exponential waves as well as to dispersion effects, which cannot be analysed in the framework of the asymptotic models commonly used for investigations of vibrations and wave propagation in the periodic shells under consideration
4.3.2 Discussion of analytical and computational results It was shown that the tolerance-periodic heterogeneity of the shells leads to exponential waves and to dispersion effects, which cannot be analysed in the framework of the asymptotic models for periodic shells
Summary
Thin linearly elastic Kirchhoff–Love-type circular cylindrical shells with a periodically micro-inhomogeneous structure (a periodically varying thickness and/or periodically varying elastic and inertial properties) in circumferential direction are analysed. We mention here monograph by Tomczyk [8], where the length-scale effect in dynamics and stability of periodic cylindrical shells is investigated, paper by Marczak and Jedrysiak [9], where vibrations of periodic three-layered plates with inert core are studied and papers by Tomczyk and Litawska [10,11], where certain extended co-called general tolerance and general asymptotic-tolerance models for the analysis of dynamic problems for periodic cylindrical shells are proposed and discussed. Periodic shells having small length dimensions are elements of air-planes, ships and machines
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