A derivation of linear elastic energies from pair-interaction atomistic systems

Pair-interaction atomistic energies may give rise, in the framework of the passage from discrete systems to continuous variational problems, to nonlinear energies with genuinely quasiconvex integrands. This phenomenon takes place even for simple harmonic interactions as shown by an example by Friesecke and Theil [19]. On the other hand, a rigorous derivation of linearly elastic energies from energies with quasiconvex integrands can be obtained by $\Gamma$-convergence following the method by Dal Maso, Negri and Percivale [14]. We show that the derivation of linear theories by $\Gamma$-convergence can be obtained directly from lattice interactions in the regime of small deformations. Our proof relies on a lower bound by comparison with the continuous result, and on a direct Taylor expansion for the upper bound. The computation is carried over for a family of lattice energies comprising interactions on the triangular lattice in dimension two.