Nonlinear homogenization of heterogeneous periodic plates of Reissner-Mindlin type
More details
Hide details
University of Lille, Department of Mathematic, Lille, French
Submission date: 2019-10-09
Acceptance date: 2019-11-06
Online publication date: 2020-04-15
Publication date: 2020-04-15
Corresponding author
Pruchnicki Erick   

Mathematic, University of Lille, France
Journal of Theoretical and Applied Mechanics 2020;58(2):317-323
In this paper, we propose a multiscale finite-strain plate theory for a composite nonlinear plate described by a repetitive periodic heterogeneity. We consider two scales, the macroscopic scale is linked to the entire plate and the microscopic scale is linked to the size of the heterogeneity. At the macroscopic scale, we approximate the displacement field by the Reissner-Mindlin model. By considering the equivalence between variations of the macroscopic elastic energy at each point of the mid surface and the microscopic one, we deduce that the macroscopic stress resultants can be expressed in terms of the microscopic stress.
Berdichevskii V., 1979, Variational asymptotic method of constructing a theory of shells, Journal of Applied Mathematics and Mechanics, 43, 4, 664–687.
Caillerie D., 1984, Thin elastic and periodic plates, Mathematical Models and Methods in Applied Sciences, 6, 1, 159-191.
Cecchi A., Sab K., 2007, A homogenized Reissner-Mindlin model for orthotropic periodic plates: application to brickwork panels, International Journal of Solids and Structures, 44, 18-19, 6055-6079.
Coenen E.W.C., Kouznetsova V.G., Geers M.G.D., 2010, Computational homogenization for heterogeneous thin sheets, International Journal for Numerical Methods in Engineering, 83, 8-9,1180-1205.
Cong Y., Nezambadi S., Zarhouni H., Yvonnet J., 2015, Multiscale computational homogenization of heterogeneous shells at small strains with extensions to finite displacements and buckling, International Journal for Numerical Methods in Engineering, 104, 4, 235-259.
Geers M.G.D., Coenen E.W.C., Kouznetsova V.G., 2007, Multi-scale computational homogenization of structured thin sheets, Modelling and Simulation in Materials Science and Engineering, 15, 4, S393-S404.
Kalamkarov A.L., Tornabene F., Pacheco P.M.C.L., Savi A., Saha G.C., 2017, Geometrically non-linear elastic model for a thin composite layer with wavy surfaces, Journal of Applied Mathematics and Mechanics, 97, 11, 1381-1392.
Kohn R.V., Vogelius M., 1984, A new model for thin plates with rapidly varying thickness, International Journal of Solids and Structures, 20, 4, 333-350.
Lebée A., Sab K., 2011, A bending-gradient model for thick plates. Part I: Theory, International Journal of Solids and Structures, 48, 20, 2878-2888.
Lebée A., Sab K., 2012, Homogenization of thick periodic plates: Application of the Bending-Gradient plate theory to a folded core sandwich panel, International Journal of Solids and Structures, 49, 19-20, 2778-2792.
Lee C.Y., Yu W., Hodges D.H., 2014, Refined modeling of composite plates with in plane heterogeneity, Journal of Applied Mathematics and Mechanics, 94, 1-2, 85-100.
Lewiński T., 1991, Effective models of composite periodic plates – 1. Asymptotic solution, International Journal of Solids and Structures, 27, 9, 1155-1172.
Petracca M., Pelà L., Rossi R., Oller S., Camata G., Spacone E., 2017, Multiscale computational first order homogenization of thick shells for the analysis of out-of-plane loaded masonry walls, Computer Methods in Applied Mechanics and Engineering, 315, 273-301.
Polizzotto C., 2018, A class of shear deformable isotropic elastic plates with parametrically variable warping shapes, ZAMM – Zeitschrift f¨ur Angewandte Mathematik und Mechanik, 98, 2, 195-221.
Pruchnicki E., 2019a, An exact two-dimensional model for heterogeneous plates, Mathematic and Mechanic of Solids, 24, 3, 637-652.
Pruchnicki E., 2019b, On the homogenization of a nonlinear shell, Mathematics and Mechanics of Solids, 24, 4, 1054-1064.
Terada K., Hirayama N., Yamamoto K., Muramatsu M., Matsubara S., Nishi S.-N., 2016, Numerical plate testing for linear two-scale analyses of composite plates with in-plane periodicity, International Journal of Numerical Methods in Enging, 105, 2, 111-137.
Schneider P., Kienzler R., Böhm M., 2014, Modeling of consistent second-order plate theories for anisotropic materials, Journal of Applied Mathematic and Mechanic, 94, 1-2, 21-42.
Sutyrin V., 1997, Derivation of plate theory accounting asymptotically correct shear deformation, Journal of Applied Mechanic, 64, 4, 905-915.
Journals System - logo
Scroll to top