# Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation

## TL;DR

This paper validates a second-gradient elastic model derived from hexagonal lattices, demonstrating its ability to capture non-local, anisotropic, and non-centrosymmetric behaviors, with potential applications in micromechanics.

## Contribution

It introduces a second-order identification method for second-gradient elastic materials from hexagonal lattices, capturing complex mechanical characteristics beyond first-order approximations.

## Key findings

- Second-order models exhibit non-locality and anisotropy.
- Validation shows good agreement between lattice responses and the SGE model.
- Higher-order models can incorporate internal length scales for advanced applications.

## Abstract

Positive definiteness and symmetry of the constitutive tensors describing a second-gradient elastic (SGE) material, which is energetically equivalent to a hexagonal planar lattice made up of axially deformable bars, are analyzed by exploiting the closed form-expressions obtained in part I of the present study in the \lq condensed' form. It is shown that, while the first-order approximation leads to an isotropic Cauchy material, a second-order identification procedure provides an equivalent model exhibiting non-locality, non-centrosymmetry, and anisotropy. The derivation of the constitutive properties for the SGE from those of the \lq condensed' one (obtained by considering a quadratic remote displacement which generates stress states satisfying equilibrium) is presented. Comparisons between the mechanical responses of the periodic lattice and of the equivalent SGE material under simple shear and uniaxial strain show the efficacy of the proposed identification procedure and therefore validate the proposed constitutive model. This model reveals that, at higher-order, a lattice material can be made equivalent to a second-gradient elastic material exhibiting an internal length, a finding which is now open for applications in micromechanics.