Effective medium theory for second-gradient elasticity with chirality

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 ${\ell }_{\mathrm{SG}}$ and ${\ell }_{\mathrm{chiral}}$, and the overall characteristic length of the domain $\mathrm{L}$. Depending on the different scale interactions between ${\ell }_{\mathrm{SG}}$, ${\ell }_{\mathrm{chiral}}$, $\ell$, and $\mathrm{L}$ 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.