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Micro-vibrations and wave propagation in biperiodic cylindrical shells

Tomczyk B., Litawska A.

Mechanics and Mechanical Engineering

2018

Vol. 22, nr 3

789--807

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.

Boundary layer phenomena in elastodynamics of functionally graded laminates

Rychlewska J., Woźniak Cz.

Archives of Mechanics

2006

Vol. 58, nr 4-5

431-444

The object of considerations are micro-laminated two-phase solids having macroscopic properties continuously varying in the direction normal to the layering (functionally graded laminates, FGL). The main question posed in this contribution is how to investigate both micro- and macro- dynamic response of the linear-elastic FGL. To answer this question, a new approximate mathematical model of FGL is proposed. This model is represented by a certain refined homogenized equation of motion for averaged displacements and a boundary-layer equation for intrinsic displacement fluctuations. Main attention is given to the investigations of near-initial and near-boundary dynamic phenomena in FGL, which are related to the specific form of initial and boundary conditions. For a periodically laminated medium the obtained results reduce to those derived in [7].