# Calculating thermoelasticity with LAMMPS

According to LAMMPS’s documentation, there are three ways to compute the thermoelastic tensor in general:

1. The Born-matrix method
2. The stress-strain method
3. The triclinic cell fluctuations that occur in an NPT simulation (seems difficult to converge)

## The Born-matrix method

The elastic stiffness tensor $C_{\alpha\beta\mu\nu}$ is related to the Born matrix tensor $C^\mathrm{B}_{\alpha\beta\mu\nu}$ by (Clavier et al., 2017; Krief & Ashkenazy, 2021)

where $\sigma^\mathrm{B}_{\alpha\beta}$ and $C^\mathrm{B}_{\alpha\beta\gamma\mu}$ are given by

In LAMMPS, the Born matrix is calculated using the compute born/matrix command.