Homogenization of an elastic material reinforced with thin rigid von Kármán ribbons

Abstract

We study the homogenization of an elastic material reinforced with periodic parallel elastic ribbons of higher rigidity. We assume that the ribbons are isotropic von Kármán plates of highly contrasting rigidity with respect to that of the matrix. We assume perfect adhesion along the common interface between the matrix and the ribbons. We derive effective energy functionals involving additional degrees of freedom implying nonlocal effects associated with thin boundary layer phenomena and additional integral terms, including macroscopic fields of extension, flexion and torsion.

Dimension reduction analysis of a three-dimensional thin elastic plate reinforced with fractal ribbons

“…The displacement w m ; m = 1, 2, which belongs to the space H 1 loc R 2+ , R 2 , is given (see for instance [12,14,29]) by…”

“…Observe that w (y) is also the solution of problem (6.5) posed in the half-plane R 2− . We now state the following preliminary result in this subsection: Proposition 14 ([12,Proposition 7]). One has 2 , where D (0, R) is a disk of radius R centred at the origin.…”

“…Starting from the pioneering work by Adkins and Rivlin [1] who studied the deformation of a structure reinforced with thin parallel, flexible and inextensible cords, strong research efforts have been devoted to the study of reinforced structures in order to describe their constitutive parameters. A lot of earlier works have focused on the homogenisation of elastic materials reinforced with fibres or ribbons composed of highly contrasting elastic materials (see for instance [5,12,15], and the references therein). The obtained homogenised composites are generally characterised by high strength and improved stiffness.…”

Asymptotic analysis of an elastic material reinforced with thin fractal strips

“…where R 2+ = y = (y 1 , y 2 ) ∈ R 2 ; y 2 > 0 and e m = (δ 1m , δ 2m ); m = 1, 2. The displacement w m ; m = 1, 2, which belongs to the space H 1 loc R 2+ , R 2 , is given (see for instance [34] and [18]) by…”

“…The homogenization of elastic materials reinforced with highly contrasted inclusions has been considered by several authors in the two last decades (see for instance [6], [10], [17], [21], [18], and the references therein). The main result is that the materials obtained by the homogenization procedure have new elastic properties.…”