Meccanica DOI 10.1007/s11012-012-9546-1 On the elastostatics of thin periodic plates with large deflections Lukasz Domagalski · Jaroslaw J˛ edrysiak Received: 4 May 2011 / Accepted: 29 February 2012 © Springer Science+Business Media B.V. 2012 Abstract Thin periodic plates with large (i.e. of the order of plate thickness) deflections are considered. In this note the tolerance and the asymptotic models of these plates are presented. As an example of applica- tions, these models are used to analyse a bending of periodic plates under various loadings. Keywords Thin periodic plate · Large deflections · Tolerance modelling 1 Preliminaries The main objects of considerations are thin linear- elastic plates with a periodic structure in planes paral- lel to the plate midplane with deflections of the order of the plate thickness, cf. Fig. 1. Plates of this kind are described by nonlinear partial differential equa- tions, which have non-continuous, highly oscillating, periodic coefficients. These equations are not a good tool to investigate special problems of these plates. Hence, various simplified models, with averaged plate properties, are proposed. Between them it can be men- tioned those based on the asymptotic homogenization, L. Domagalski · J. J˛ edrysiak ( ) Department of Structural Mechanics, Technical University of Lód´ z, al. Politechniki 6, 90-924, Lód´ z, Poland e-mail: jarek@p.lodz.pl L. Domagalski e-mail: l.domagalski@gmail.com cf. Kohn and Vogelius [11]. However, in equations of these models the effect of the microstructure size on the plate behaviour is usually neglected. On the other hand, elastostatic problems of thin plates under large deflections are described by the known geometrically nonlinear equations, which are presented in e.g. Tim- oshenko and Woinowsky-Krieger [17], Wo´ zniak (ed.) [19]. Using equations of the three-dimensional nonlin- ear continuum mechanics there are derived equations of von Kármán-type plate theories by Meenen and Al- tenbach [14]. To investigate bending problems of these plates various methods can be used, cf. Levy [13], Timoshenko and Woinowsky-Krieger [17], Huang Jia- yin [4], Chia [2]. In order to describe the effect of the microstructure size in periodic plates with large deflections, the tol- erance modelling is applied, cf. the books edited by Wo´ zniak, Michalak and J˛ edrysiak [21] and by Wo´ z- niak et al. [20]. Applications of this method to var- ious periodic structures are shown in a series of pa- pers, e.g. for vibrations of periodic wavy-type plates by Michalak [15], for periodically stiffened plates by Nagórko and Wo´ zniak [16], for the buckling of peri- odic thin plates by J˛ edrysiak [5], for plates with the inhomogeneity period of an order of the plate thick- ness by Baron [1], for multiperiodic reinforced com- posites by J˛ edrysiak and Wo´ zniak [9], for stability analysis of periodic shells by Tomczyk [18], for sta- bility and vibrations of periodic plates by J˛ edrysiak [6, 7], for some problems of bending of thin periodic plates by Domagalski and J˛ edrysiak [3], for vibrations