General up to next-nearest neighbour elasticity of triangular lattices in three dimensions

TL;DR

This paper derives the most general form of discrete elasticity for 2D triangular lattices in three dimensions, considering up to next-nearest neighbor interactions, with implications for physics and materials science.

Contribution

It provides a comprehensive elasticity model for 2D triangular lattices in 3D, including rotational invariance and edge effects, extending previous models.

Findings

Identifies the necessary terms for rotational invariance.

Includes edge vertex elasticity explicitly.

Clarifies the validity of previously suspected terms.

Abstract

We establish the most general form of the discrete elasticity of a 2D triangular lattice embedded in three dimensions, taking into account up to next-nearest neighbour interactions. Besides crystalline system, this is relevant to biological physics (e.g., red blood cell cytoskeleton) and soft matter (e.g., percolating gels, etc.). In order to correctly impose the rotational invariance of the bulk terms, it turns out to be necessary to take into account explicitly the elasticity associated with the vertices located at the edges of the lattice. We find that some terms that were suspected in the litterature to violate rotational symmetry are in fact admissible