Dynamics of medium thickness plates interacting wi

Authors: Jarosław Jędrysiak Agnieszka Paś

Publish Date: 2014/04/10

Volume: 49, Issue: 7, Pages: 1577-1585

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Abstract

Medium thickness plates resting on a periodic Winkler’s foundation are investigated New averaged nonasymptotic models for those plates are proposed These models are based on the tolerance averaging technique The main feature of these models is that they describe the effect of period lengths on the overall behaviour of the plate It is also shown that from governing equations of these models equations of simplified averaged models called asymptotic models can be obtained An additional interesting feature of the proposed models is that the equations describe also the effect of normal stress in the thickness directionPlates interacting with a subsoil are often applied as elements of constructions in the civil engineering eg as elements of building foundations or reinforcements of roads foundations In many cases as a certain approximation the subsoil is modelled as a Winkler’s foundationThe above systems ie plates on a periodic foundation consist of many small identical elements called periodicity cells Properties of these structures are described by highly oscillating periodic and often noncontinuous functions Because an analysis of engineering problems of these plates is too complicated using exact equations of the plate theory different averaged models have been proposed These models have usually described certain homogeneous plates with constant homogenized properties instead of real periodic plates Between these models it can be mentioned those based on the method of asymptotic homogenization for periodic solids proposed in Bensoussan et al 3 Models of this kind for periodic plates were presented in a series of papers eg Caillerie 5 Kohn and Vogelius 17 Other models of these plates are based on the microlocal parameters approach cf Matysiak and Nagórko 18 However the aforementioned models usually neglect the effect of period lengths on the overall dynamic plate behaviour Behaviour of functionally graded plates resting on a foundation is also analysed eg Tahouneh and Naei 21 where considerations are based on the threedimensional elasticity theory Yajuvindra Kumar and Lal 26 where vibrations of nonhomogeneous plates with varying thickness interacting with a foundation are investigatedIn order to take into account this effect new nonasymptotic averaged models based on the tolerance averaging technique have been proposed This approach was discussed for periodic composites and structures in the monograph by Woźniak and Wierzbicki 25 and in the book edited by Woźniak et al 24 The tolerance averaging procedure were applied to analyse nonstationary problems for different periodic structures in many papers eg for periodic grounds by Dell’Isola et al 7 for wavy plates by Michalak 19 Kirchhoff plates by Jędrysiak 11 12 13 14 for Hencky–Bolle plates by Baron 2 for honeycomb latticetype plates by Cielecka and Jędrysiak 6 for thin cylindrical shells Tomczyk 22 These papers showed that the effect of period lengths plays a crucial role in dynamics of periodic structures Moreover some static problems of periodic thin plates with moderately large deflections were analysed by Domagalski and Jędrysiak 8 This modelling method was also applied to analysis some dynamical problems of functionally graded media or structures eg transversally graded thin plates were considered by Kaźmierczak and Jędrysiak 16 and longitudinally graded thin plates by Michalak and Wirowski 20The main aim of this note is to formulate a new nonasymptotic averaged model which describes the above effect on nonstationary problems of medium thickness plates interacting with a periodic Winkler’s foundation The peculiar feature of the proposed model is that the plane stress assumption S 33 = 0 is omitted ie the effect of the stress S 33 is not neglected cf Jemielita 10Considerations of this contribution are based on the wellknown Hencky–Bolle plate theory assumptions cf Bolle 4 Hencky 9 which are extended on the effect of Winkler’s foundation cf Ambartsumyan 1 and the effect of the stress S 33 cf Jemielita 10Let us denote by 0x 1 x 2 x 3 the orthogonal Cartesian coordinate system in the physical space and by t the time coordinate Let subscripts α β …i j … run over 1 2 over 1 2 3 and indices A B… a b… run over 1… N 1… n Summation convention holds for all aforementioned indices Denote also x ≡ x 1 x 2 and z ≡ x 3 Let us assume that the undeformed plate occupies the region Ω ≡ x z−d/2  z  d/2 x ∈ Π where Π is the midplane with length dimensions L 1 L 2 along the x 1 and x 2axis respectively and d is the plate thickness

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