HOMOGENIZATION OF PERIODIC GRAPH-BASED ELASTIC STRUCTURES

H Abdoul-Anziz (1, 2) , Pierre Seppecher (2, 1)

1 IMATH - Institut de Mathématiques de Toulon - EA 2134
2 LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]

In the framework of Γ-convergence and periodic homogenization of highly contrasted materials, we study cylindrical structures made of one material and voids. Interest in high contrast homogenization is growing rapidly but assumptions are generally made in order to remain in the framework of classical elasticity. On the contrary, we obtain homogenized energies taking into account second gradient (i.e. strain gradient) effects. We first show that we can reduce the study of the considered structures to discrete systems corresponding to frame lattices. Our study of such lattices differs from the literature in the fact that we must take into account the different orders of magnitude of the extensional and flexural stiffnesses. This allows us to consider structures which would have been floppy when considering only extensional stiffness and completely rigid when considering flexural stiffnesses of the same order of magnitude than the extensional ones. At our knowledge, this paper provides the first rigorous homogenization result with a complete second gradient limit energy.

Mots clés: Γ-convergence; periodic homogenization; high contrast

Domaines: Mathématiques [math]; Mécanique des solides [physics.class-ph]; Matériaux et structures en mécanique [physics.class-ph]

H Abdoul-Anziz, Pierre Seppecher. HOMOGENIZATION OF PERIODIC GRAPH-BASED ELASTIC STRUCTURES. 2017. ⟨hal-01492589v2⟩