Mechanical stability conditions for 3D and 2D crystals under arbitrary load

TL;DR

This paper unifies mechanical stability conditions for 3D and 2D crystals under any load using spectral decomposition and Kelvin moduli, providing explicit criteria and a computational tool for stability analysis across symmetry classes.

Contribution

It introduces a comprehensive framework for assessing stability of crystals under arbitrary loads, including explicit conditions and a Mathematica tool for diverse symmetries.

Findings

Unified stability conditions for all crystal symmetries.

Explicit criteria for higher symmetry crystals.

A computational tool for stability analysis.

Abstract

The paper gathers and unifies mechanical stability conditions for all symmetry classes of 3D and 2D materials under arbitrary load. The methodology is based on the spectral decomposition of the fourth-order stiffness tensors mapped to second-order tensors using orthonormal (Mandel) notation, and the verification of the positivity of the so-called Kelvin moduli. An explicit set of stability conditions for 3D and 2D crystals of higher symmetry is also included, as well as a Mathematica notebook that allows mechanical stability analysis for crystals, stress-free and stressed, of arbitrary symmetry under arbitrary loads.