  We review and compare the Born-Huang and the Lemaitre-Maloney's theories that
lead to analytical expressions for elastic constants, accounting for affine and
nonaffine deformations in a lattice. The Born-Huang method is based on
Helmholtz energy while the Lemaitre-Maloney's formalism focus on Gibbs force.
Although starting from different perspectives, in the linear elastic limit, and
in equilibrium, elastic material constants must be the same in all these
methods. This is explicitly verified on examples of linear chains, and
numerical simulation of a non-centrosymmetric crystal.
