Plates with incompatible prestrain of higher order

• Published in: arXiv.org
• Main Authors: Lewicka, Marta, Raoult, Annie, Ricciotti, Diego
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 30.03.2015
• Subjects: Elasticity; Scaling; Thin plates
• ISSN: 2331-8422

We study the effective elastic behaviour of the incompatibly prestrained thin plates, characterized by a Riemann metric \(G\) on the reference configuration. We assume that the prestrain is "weak", i.e. it induces scaling of the incompatible elastic energy \(E^h\) of order less than \(h^2\) in terms of the plate's thickness \(h\). We essentially prove two results. First, we establish the \(\Gamma\)-limit of the scaled energies \(h^{-4}E^h\) and show that it consists of a von Kármán-like energy, given in terms of the first order infinitesimal isometries and of the admissible strains on the surface isometrically immersing \(G_{2\times 2}\) (i.e. the prestrain metric on the midplate) in \(\mathbb{R}^3\). Second, we prove that in the scaling regime \(E^h\sim h^\beta\) with \(\beta>2\), there is no other limiting theory: if \(\inf h^{-2} E^h \to 0\) then \(\inf E^h\leq Ch^4\), and if \(\inf h^{-4}E^h\to 0\) then \(G\) is realizable and hence \(\min E^h = 0\) for every \(h\).