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On the combined asymptotic-tolerance modelling of dynamic problems for thin biperiodic cylindrical shells

Tomczyk B., Litawska A.

Vibrations in Physical Systems

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2018

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Vol. 29

art. no. 2018020

EN

The objects of consideration are thin linearly elastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss a new averaged general asymptotic-tolerance model for the analysis of selected dynamic problems for the shells under consideration. This model is derived by applying the combined modelling which includes two techniques: the asymptotic modelling procedure and a certain extended version of the known tolerance non-asymptotic modelling technique based on a new notion of weakly slowly-varying function. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, governing equations of the averaged combined model have constant coefficients depending also on a cell size. The differences between the general combined model proposed here and the corresponding known standard combined model derived by means of the more restrictive concept of slowly-varying functions are discussed.

2

Micro-vibrations and wave propagation in biperiodic cylindrical shells

Tomczyk B., Litawska A.

Mechanics and Mechanical Engineering

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2018

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Vol. 22, nr 3

789--807

EN

The objects of consideration are thin linearly elastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to study a certain long wave propagation problem related to micro-fluctuations of displacement field caused by a periodic structure of the shells. This micro-dynamic problem will be analysed in the framework of a certain mathematical averaged model derived by means of the combined modelling procedure. The combined modelling applied here includes two techniques: the asymptotic modelling procedure and a certain extended version of the known tolerance non-asymptotic modelling technique based on a new notion of weakly slowly-varying function. Both these procedures 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 averaged combined model have constant coefficients depending also on a cell size. It will be shown that the micro-periodic heterogeneity of the shells leads to exponential 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.

3

A new tolerance model of vibrations of thin microperiodic cylindrical shells

Tomczyk B., Litawska A.

Czasopismo Inżynierii Lądowej, Środowiska i Architektury / Journal of Civil Engineering, Environment and Architecture

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2017

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z. 64, nr 2/I

203--216

EN

The objects of consideration are thin linearly elastic Kirchhoff-Love-type circular cylindrical shells having a micro-periodic structure in circumferential direction (uniperiodic shells). At the same time the shells have constant structure in axial direction. The aim of this contribution is to formulate and discuss a new nonasymptotic averaged model for the analysis of selected dynamic problems for these shells. This, so-called, general tolerance model is derived by means of a certain extended version of the known tolerance modelling of micro-heterogeneous media. This version is based on a new notion of weakly slowly-varying functions. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, governing equations of the tolerance model have constant coefficients depending also on a period of inhomogeneity. Hence, the model makes it possible to investigate the effect of a cell size on the global shell dynamics (the length-scale effect). The differences between the general tolerance model proposed here and the corresponding known standard tolerance model derived by means of the more restrictive concept of slowly-varying functions are discussed.

PL

Przedmiotem rozważań są cienkie liniowo-sprężyste powłoki walcowe typu Kirchhoffa- Love’a mające periodycznie mikro-niejednorodną strukturę w kierunku obwodowym. Powłoki takie nazywamy uniperiodycznymi. Celem pracy jest sformułowanie nowego, nieasymptotycznego, uśrednionego modelu służącego do analizy wybranych zagadnień dynamiki takich powłok. Przedstawiony ogólny model tolerancyjny wyprowadzony jest w oparciu o pewną zmodyfikowaną wersję znanej techniki tolerancyjnego modelowania struktur mikro-niejednorodnych. Wersja ta bazuje na nowym pojęciu funkcji słabo wolno-zmiennej. W przeciwieństwie do równań wyjściowych dla analizowanych powłok niejednorodnych mających współczynniki periodyczne, silnie oscylujące i nieciągłe, równania modelu tolerancyjnego mają stałe współczynniki. Ponadto, współczynniki te zależą od parametru długości mikrostruktury. Tym samym umożliwiają badanie efektu skali.