# Identification of second-gradient elastic materials from planar hexagonal lattices. Part I: Analytical derivation of equivalent constitutive tensors

## TL;DR

This paper derives analytical expressions for the non-local constitutive tensors of a second-gradient elastic material modeled as a homogeneous solid equivalent to a hexagonal lattice, linking lattice geometry to continuum properties.

## Contribution

It provides a closed-form analytical derivation of the equivalent second-gradient elastic tensors from a specific hexagonal lattice model, considering energy equivalence and remote displacement constraints.

## Key findings

- Explicit formulas for non-local elastic parameters
- Reduction to classical elasticity in the limit of small unit cells
- Framework for numerical validation of the derived tensors

## Abstract

A second-gradient elastic (SGE) material is identified as the homogeneous solid equivalent to a periodic planar lattice characterized by a hexagonal unit cell, which is made up of three different linear elastic bars ordered in a way that the hexagonal symmetry is preserved and hinged at each node, so that the lattice bars are subject to pure axial strain while bending is excluded. Closed form-expressions for the identified non-local constitutive parameters are obtained by imposing the elastic energy equivalence between the lattice and the continuum solid, under remote displacement conditions having a dominant quadratic component. In order to generate equilibrated stresses, in the absence of body forces, the applied remote displacement has to be constrained, thus leading to the identification in a \lq condensed' form of a higher-order solid, so that imposition of further constraints becomes necessary to fully quantify the equivalent continuum. The identified SGE material reduces to an equivalent Cauchy material only in the limit of vanishing side length of hexagonal unit cell. The analysis of positive definiteness and symmetry of the equivalent constitutive tensors, the derivation of the second-gradient elastic properties from those of the higher-order solid in the \lq condensed' definition, and a numerical validation of the identification scheme are deferred to Part II of this study.