Article Mathematics and Mechanics of Solids 1–16 Ó The Author(s) 2018 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1081286517752544 journals.sagepub.com/home/mms An exact two-dimensional model for heterogeneous plates E Pruchnicki Department of Mathematics, University of Lille, F 59655 Villeneuve d’Ascq, France Received 18 September 2017; accepted 15 December 2017 Abstract This work derives an exact two-dimensional plate theory for heterogeneous plates consistent with the principle of sta- tionary three-dimensional potential energy under general loading. We do not take any hypotheses about the shape of the heterogeneity. We start from three-dimensional linear elasticity and by using the Fourier series expansion in the thickness direction of the displacement field with respect to a basis of scaled Legendre polynomials. We deduce an exact two-dimensional model expressed in power-series in the ratio between the thickness of the plate and a characteristic measurement of its mid-plane. Then we can derive an approximative theory by neglecting in the expression of potential energy all terms that contain a powerof this ratio that is higher than a given truncation power for getting to an approxi- mative two-dimensional problem. In the last section, we show that the solution of the approximation problem only dif- fers from the exact solution by a difference of the same orderof the neglected terms in the potential energy. A similar result when we truncate the displacement field can be also established. This model can be a starting point to formulate a two-dimensional homogenized boundary value problem for highly heterogeneous periodic plates. Keywords Linear elasticity, plate theory, heterogeneous plate, Fourier series expansion, scaled Legendre polynomials, rigorous con- vergence result. 1. Introduction Plates are very important engineering structures, which have attracted extensive research since the 19th century. The subject of this work is a statical analysis of a linearly elastic heterogeneous plate. A thin plate is a three-dimensional body with a small transverse dimension, called the thickness, compared to the two other dimensions of the mid-surface. The aim of plate theory is to attempt to reduce the three- dimensional elasticity theory to a two-dimensional exact and then approximate one defined on a sur- face. A common starting point is the series expansion of the vector displacement field deformed in terms of thickness h. Some consistent mathematical approaches for deriving leading-order plate theories are based on certain a priori scalings between the thickness and the deformations or applied loads. In this vein, asymptotic methods aim at generating the leading-order two-dimensional variational problem by postulating formal asymptotic expansion of the displacement field. A mathematical background of the asymptotic approach in the context of both linear and nonlinear elasticity is given by Ciarlet [1]. These theories generate a hierarchy of the two-dimensional model depending on the order of magnitude of the applied load with respect to the thickness of the plate [2]. However, in reality applied loads are external and thus are not linked to the geometry of the plate. In general, as remarked upon by Friesecke et al. Corresponding author: E Pruchnicki, Department of Mathematics, University of Lille, F 59655 Villeneuve d’Ascq, France. Email: erick.pruchnicki@gmail.com