The authors of this interesting paper investigate the process of deformation under bending of nonlinearly elastic plates of very small thickness tending to zero. The material of the plate has thickness \(h\to 0\) and possesses an \(\varepsilon \)-periodic structure such that \(\varepsilon^{-2}h\to 0\). It turns out that the plates exhibit non-standard behaviour in the asymptotic two-dimensional reduction from three-dimensional elasticity. There is an observation that in general, their effective stored-energy density is “discontinuously anisotropic” in all directions.
The first ideas and methods to study these objects belong to E. Acerbi et al. [J. Elasticity 25, No. 2, 137–148 (1991; Zbl 0734.73094)], H. Le Dret and A. Raoult [J. Math. Pures Appl. (9) 74, No. 6, 549–578 (1995; Zbl 0847.73025)] and were continued by O. Pantz [“Quelques problèmes de modélisation en élasticité non linéaire”, Thèse de l’Université Pierre et Marie Curie (2001)], G. Friesecke et al. [Commun. Pure Appl. Math. 55, No. 11, 1461–1506 (2002; Zbl 1021.74024); Arch. Ration. Mech. Anal. 180, No. 2, 183–236 (2006; Zbl 1100.74039)]. The observation here is that the homogeneous plates of thickness \(h\), represented as three-dimensional nonlinearly elastic bodies, afford a special compactness argument for sequences of deformation gradients \(\nabla u\) with elastic energy of order \(O(h^3 )\) as \(h\to 0\). It is introduced a new “rigidity estimate” concerning the distance of local values of \(\nabla u\) from the group of rotations SO(3). Consider the “limit” elastic energy functional as \(h\to 0\), \[ E_{\mathrm{lim}}(u)=12^{-1}\int_{\omega } Q_2(\Pi (u)), \;\;u\in H_{\mathrm{iso}}^{2}(\omega), \] where \(\omega\subset\mathbb{R}^2\) is the mid-surface of the undeformed plate, \(\Pi (u)=(\nabla'u)^T\nabla' (\partial_1u\bigwedge\partial_2u)\) (\(\nabla'=(\partial_1,\partial_2)\)) is the matrix of the second fundamental form of an isometric surface \(u:\omega\to \mathbb{R}^{3}\), and \(Q_2\) is a quadratic form derived via dimension reduction from a quadratic form appearing in the process of linearisation of elastic properties of the material in the small-strain regime. Considering real-world materials the authors pose the question: “in what way the above result is affected by a possible inhomogeneity of material properties of the plate in the directions tangential to its mid-surface.”
The main result here concerns the supercritical regime, that is, \(h\ll \varepsilon^2\). The limit energy functional has the form \[ E_{\mathrm{hom}}^{sc}=(12)^{-1}\int_{\omega} Q_{\mathrm{hom}}^{sc}(\Pi (u)), \] where \(Q_{\mathrm{hom}}^{sc}=\min\int_{Y} Q_2(y,\Pi + \nabla_{y}^2\psi )dy\), \(\psi \in H_{\mathrm{loc}}^{2}(\mathbb{R}^2)\) is \(Y\)-periodic, subject to the isometry constraint equality \(\det{(\Pi + \nabla_{y}^2\psi )}=0\). Here \(\psi = \psi (x',y)\), \(x'\in\omega \), \(y\in Y\equiv [0,1)^2\) is a term from the two-scale limit of the original sequence of deformations. Note that the behavior of the plate on \(\varepsilon \)-scale depends on this term. The above stated isometry constraint implies that \(Q^{sc}(\Pi )\) is “discontinuously anisotropic in all directions of bending as a function of the macroscopic deformation gradient \(\nabla u\)”. The authors claim that probably one may encounters here with a new phenomenon for nonlinearly elastic plates. Finally, they conclude that the analysis of the “critical” scaling \(h \sim \varepsilon^2\) is currently open.