Plates with incompatible prestrain of high 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 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 Γ-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_{2×2} (i.e. the prestrain metric on the midplate) in R^3. Second, we prove that in the scaling regime E^h~ h^β with β > 2, there is no other limiting theory: if inf h^{−2} E^h → 0 then inf E^h ≤ Ch^4 , and if inf h^{−4} E^h → 0 then G is realizable and hence min E^h = 0 for every h.
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Annales de l'Institut Henri Poincaré, 2017, 34, pp.1883-1912. 〈10.1016/j.anihpc.2017.01.003 〉
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Marta Lewicka, Annie Raoult, Diego Ricciotti. Plates with incompatible prestrain of high order. Annales de l'Institut Henri Poincaré, 2017, 34, pp.1883-1912. 〈10.1016/j.anihpc.2017.01.003 〉. 〈hal-01138338〉

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