Plates with incompatible prestrain of higher order
Abstract
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 Eh of order less than h2 in terms of the plate's thickness h. We essentially prove two results. First, we establish the -limit of the scaled energies h-4Eh and show that it consists of a von K\'arm\'an-like energy, given in terms of the first order infinitesimal isometries and of the admissible strains on the surface isometrically immersing G2× 2 (i.e. the prestrain metric on the midplate) in R3. Second, we prove that in the scaling regime Eh hβ with β>2, there is no other limiting theory: if ∈f h-2 Eh 0 then ∈f Eh≤ Ch4, and if ∈f h-4Eh 0 then G is realizable and hence Eh = 0 for every h.