# A nonlinear hyperelasticity model for single layer blue phosphorus based on ab-initio calculations

## TL;DR

This paper introduces a fully nonlinear hyperelasticity model for single layer blue phosphorus, calibrated from ab-initio DFT calculations, capturing anisotropic large-strain behavior and validated through simulations.

## Contribution

The paper develops the first fully nonlinear hyperelastic model for blue phosphorus based on ab-initio data, enabling accurate large-strain simulations.

## Key findings

- Model shows good agreement with DFT results.
- Can be integrated into shell finite element methods.
- Elasticity constants at small deformations are provided.

## Abstract

A new hyperelastic membrane material model is proposed for single layer blue phosphorus ($\beta\text{-P}$), also known as blue phosphorene. The model is fully nonlinear and captures the anisotropy of $\beta\text{-P}$ at large strains. The material model is calibrated from density functional theory (DFT) calculations considering a set of elementary deformation states. Those are pure dilatation and uniaxial stretching along the armchair and zigzag directions. The material model is compared and validated with additional DFT results and existing DFT results from the literature, and the comparison shows good agreement. The new material model can be directly used within computational shell formulations that are for example based on rotation-free isogeometric finite elements. This is demonstrated by simulations of the indentation and vibration of single layer blue phosphorus sheets at micrometer scales. The elasticity constants at small deformations are also reported.