# Calculating thermoelasticity with LAMMPS

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

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

## The Born-matrix method

The elastic stiffness tensor is related to the Born matrix tensor by (Clavier et al., 2017; Krief & Ashkenazy, 2021)

where and are given by

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

command.

## References

- Clavier, G., Desbiens, N., Bourasseau, E., Lachet, V., Brusselle-Dupend, N., & Rousseau, B. (2017). Computation of elastic constants of solids using molecular simulation: Comparison of constant volume and constant pressure ensemble methods.
*Molecular Simulation*,*43*(17), 1413–1422. https://doi.org/10.1080/08927022.2017.1313418 - Krief, M., & Ashkenazy, Y. (2021). Calculation of elastic constants of embedded-atom-model potentials in the N V T ensemble.
*Physical Review E*,*103*(6), 063307. https://doi.org/10.1103/PhysRevE.103.063307 - Lutsko, J. F. (1989). Generalized expressions for the calculation of elastic constants by computer simulation.
*Journal of Applied Physics*,*65*(8), 2991–2997. https://doi.org/10.1063/1.342716 - Zhen, Y., & Chu, C. (2012). A deformation–fluctuation hybrid method for fast evaluation of elastic constants with many-body potentials.
*Computer Physics Communications*,*183*(2), 261–265. https://doi.org/10.1016/j.cpc.2011.09.006 - 8.3.4. Calculate elastic constants — LAMMPS documentation
- lammps/examples/ELASTIC_T/BORN_MATRIX/Silicon at develop · lammps/lammps (github.com)