Effective medium theory for second-gradient elasticity with chirality

TL;DR

This paper develops effective models for heterogeneous second-gradient elastic materials with chirality, considering multiple characteristic lengths to determine when a Cauchy or second-gradient continuum model applies.

Contribution

It introduces a classification of effective equations based on the hierarchy of four characteristic lengths in chiral, second-gradient elastic materials.

Findings

Effective models depend on the scale interactions between heterogeneity size, constituent lengths, and domain size.

The methodology combines scaling arguments with periodic homogenization and unfolding techniques.

The work provides criteria for when to use Cauchy versus second-gradient continuum models.

Abstract

We derive effective, parsimonious models from a heterogeneous second-gradient nonlinear elastic material taking into account chiral scale-size effects. Our classification of the effective equations depends on the hierarchy of four characteristic lengths: The size of the heterogeneities $\ell$, the intrinsic lengths of the constituents $\ell_{\rm SG}$ and $\ell_{\rm chiral}$, and the overall characteristic length of the domain ${\rm L}$. Depending on the different scale interactions between $\ell_{\rm SG}$, $\ell_{\rm chiral}$, $\ell$, and ${\rm L}$ we obtain either an effective Cauchy continuum or an effective second-gradient continuum. The working technique combines scaling arguments with the periodic homogenization asymptotic procedure. Both the passage to the homogenization limit and the unveiling of the correctors' structure rely on a suitable use of the periodic unfolding and related operators.