Plates with incompatible prestrain of high order

Lewicka, Marta; Raoult, Annie; Ricciotti, Diego

doi:10.1016/j.anihpc.2017.01.003

Lewicka, Marta ; Raoult, Annie ; Ricciotti, Diego

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1883-1912.

Résumé
Abstract

On s'intéresse au comportement de structures minces d'épaisseur h dont l'énergie interne $E^{h}$ est régie par une métrique riemannienne tridimensionnelle G imposée, constante dans l'épaisseur, n'admettant pas nécessairement d'immersion isométrique. On sait que lorsque la restriction de G à la surface moyenne ω possède une immersion isométrique suffisamment régulière, c'est-à-dire appartenant à $W^{2, 2} (ω, R^{3})$ , alors $h^{- 2} \inf E^{h}$ admet une limite finie c quand h tend vers 0. Le modèle limite correspondant généralise le modèle de flexion non linéaire, classique pour la métrique euclidienne. Nous nous plaçons ici dans le cas où c vaut 0, ce qui équivaut à la nullité de trois des six coeffiecients du tenseur de courbure associé à G. Nous montrons qu'alors $\inf E^{h} \leq C h^{4}$ . Nous identifions la Γ-limite de $h^{- 4} E^{h}$ et montrons qu'elle généralise l'énergie de von Kármán. Elle s'exprime en fonction des déplacements incrémentaux par rapport à la surface définie par le modèle de flexion et de déformations tangentielles généralisées. De plus, nous montrons que l'infimum de ce modèle limite à l'ordre 4 n'est nul que si G admet une immersion isométrique, auquel cas $\min E^{h} = 0$ pour tout h.

We study the elastic behaviour of incompatibly prestrained thin plates of thickness h whose internal energy $E^{h}$ is governed by an imposed three-dimensional smooth Riemann metric G only depending on the variable in the midsurface ω. It is already known that $h^{- 2} \inf E^{h}$ converges to a finite value c when the metric G restricted to the midsurface has a sufficiently regular immersion, namely $W^{2, 2} (ω, R^{3})$ . The obtained limit model generalizes the bending (Kirhchoff) model of Euclidean elasticity. In the present paper, we deal with the case when c equals 0. Then, equivalently, three independent entries of the three-dimensional Riemann curvature tensor associated with G are null. We prove that, in such regime, necessarily $\inf E^{h} \leq C h^{4}$ . We identify the Γ-limit of the scaled energies $h^{- 4} E^{h}$ and show that it consists of a von Kármán-like energy. The unknowns in this energy are the first order incremental displacements with respect to the deformation defined by the bending model and the second order tangential strains. In addition, we prove that when $\inf h^{- 4} E^{h} \to 0$ , then G is realizable and hence $\min E^{h} = 0$ for every h.

MR Zbl | 1 citation dans Numdam

DOI : 10.1016/j.anihpc.2017.01.003

Mots-clés : Non-Euclidean plates, Prestrained energy, Nonlinear elasticity, Calculus of variations

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     author = {Lewicka, Marta and Raoult, Annie and Ricciotti, Diego},
     title = {Plates with incompatible prestrain of high order},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1883--1912},
     publisher = {Elsevier},
     volume = {34},
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Lewicka, Marta; Raoult, Annie; Ricciotti, Diego. Plates with incompatible prestrain of high order. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1883-1912. doi : 10.1016/j.anihpc.2017.01.003. http://www.numdam.org/articles/10.1016/j.anihpc.2017.01.003/

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