© 2020 Elsevier Ltd We review and compare the Born-Huang and the Lemaitre-Maloney theories that lead to analytical expressions for elastic constants, accounting for affine and nonaffine deformations in a lattice (or in a disordered solid). The Born-Huang method is based on Helmholtz free energy while the Lemaitre-Maloney formalism focus on the Gibbs ensemble with the focus on local force. Although starting from different perspectives, in the linear elastic limit, and in equilibrium, material elastic constants must be the same in all these methods. This is explicitly verified on examples of linear chains, and the numerical simulation of a non-centrosymmetric crystal.