Plates with incompatible prestrain of high order

Abstract

Accepté dans Ann. Inst. H. Poincaré Anal. Non LinéaireInternational audienceWe 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

On the Role of Curvature in the Elastic Energy of Non-Euclidean Thin Bodies

“…Part 1 of the Theorem 1.2 is merely a restatement of a corollary of the main result of [KS14], which we include for completeness. Parts 2 and 3 generalize the conditions for a scaling of o(h 2 ) in [BLS16,LRR]; they clarify the geometric implications of this scaling also in the plate case. These are proved by carefully analyzing the limit functional obtained in [KS14].…”

“…In this paper we generalize the relations between curvature and energy scaling of thin plates [BLS16,LRR], to every dimension and co-dimension. Our results provide a unifying ground for most of the results mentioned above.…”

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“…Thirdly, in the context of the non-Euclidean energy (1.10), it has been shown in [7] that the scaling: inf 1 h E(·, Ω h ) ∼ h 2 only occurs when the metric G 2×2 on the mid-plate U can be isometrically immersed in R 3 with the regularity W 2,2 and when, at the same time, the three appropriate Riemann curvatures of G do not vanish identically; the relevant residual theory, obtained through Γ-convergence, yielded then a Kirchhoff-like residual energy. Further, in [33] the authors proved that the only outstanding nontrivial residual theory is a von Kàrmànlike energy, valid when: inf 1 h E(·, Ω h ) ∼ h 4 . This scale separation, contrary to [13,32], is due to the fact that while the magnitude of external forces is adjustable at will, it seems not to be the case for the interior mechanism of a given metric G which does not depend on h. In fact, it is the curvature tensor of G which induces the nontrivial stresses in the thin film and it has only six independent components, namely the six sectional curvatures created out of the three principal directions, which further fall into two categories: including or excluding the thin direction variable.…”

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