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

Abstract

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

Microscopic Validation of a Variational Model of Epitaxially Strained Crystalline Films

“…Rigorous Γ-convergence results [7] on the derivation of linear-elastic theories from nonlinear elastic discrete energies have been obtained in [5,22] without the presence of surface energies.…”

“…as shown in [2], must be imposed on the discrete displacements y for every mutual nearest neighbors x,x,x ∈ L ε (Ω h ), in order to avoid local reflection of the reference lattice, which might be otherwise energetically convenient (see [5]). Without such condition rigidity would fail and a more complex description of the Γ-limit would be needed that do not find currently any examples in the literature to the best of our knowledge.…”

Analytical treatment for the asymptotic analysis of microscopic impenetrability constraints for atomistic systems

“…The other two issues present in the one-dimensional analysis; i.e., the characterization of the bulk energy close to ground states and surface relaxation have been separately addressed in slightly different hypotheses by Braides, Solci and Vitali [12] (for the bulk analysis) and Theil [25] (for the external surface relaxation). ν Figure 1: A reference configuration with fracture site and its macroscopic normals, and its underlying triangulation In order to make the analysis clearer we will scale the energies by an additive constant as…”

Динамический Фазовый Переход В Простейшей Модели Цепочки Молекул

“…Однако логично исследовать также динамику системы частиц во времени и получить хорошие оценки для функционала Френкеля-Конторовой [5]. Следует отметить также работы [6]- [9], берущие начало от монографии [10], где многомерные статические модели рассматривались с целью вывода уравнений линейной упругости из микромоделей.…”

“…= const • l const • F N (l).Здесь в первом неравенстве мы воспользовались утверждением 3 леммы 3, во втором -соотношением (22), а в последнем -определением(6) функции F N (x). Объединяя полученные выкладки, выводим оценку сверху sup t 0 |I N,k,l (t)| sup t 0 |J 1 N,k,l (t)| + sup t 0 |J 2 N,k,l (t)| const • F N (l).…”