Asymptotic analysis of deformation behavior in high-contrast fiber-reinforced materials: Rigidity and anisotropy

Abstract

We identify the restricted class of attainable effective deformations in a model of reinforced composites with parallel, long, and fully rigid fibers embedded in an elastic body. In mathematical terms, we characterize the weak limits of sequences of Sobolev maps whose gradients on the fibers lie in the set of rotations. These limits are determined by an anisotropic constraint in the sense that they locally preserve length in the fiber direction. Our proof of the necessity emerges as a natural generalization and modification of the recently established asymptotic rigidity analysis for composites with layered reinforcements. However, the construction of approximating sequences is more delicate here due to the higher flexibility and connectedness of the soft material component. We overcome these technical challenges by a careful approximation of the identity that is constant on the rigid components, combined with a lifting in fiber bundles for Sobolev functions. The results are illustrated with several examples of attainable effective deformations. If an additional partial second-order regularization is introduced into the material model, only rigid body motions can occur macroscopically.

A variational perspective on auxetic metamaterials of checkerboard-type

“…Our main result (Theorem 1.1) is a homogenization result that is nonstandard compared to classical papers like [9,45] and the works on high-contrast media [13,14,21,22]. Instead, it can be interpreted in the context of asymptotic rigidity statements for other reinforcing elements like layers [15,16,20] and fibers [25]. This means, generally speaking, that the models are governed by an interesting interplay between the specific geometric pattern of the heterogeneities and a strong contrast in the elasticity constants, which leads to global effects and overall, to a strongly restricted macroscopic material response.…”

“…They are both embedded in a general proof strategy of asymptotic rigidity results (cf. [16,20,25] and also [27]) essential for expanding the observations on the asymptotic behavior of sequences (u ε ) ε ⊂ A when the exact differential inclusion (1.12) is weakened to the approximate version…”