Connecting Atomistic and Continuous Models of Elastodynamics

Abstract

We prove long-time existence of solutions for the equations of atomistic elastodynamics on a bounded domain with time-dependent boundary values as well as their convergence to a solution of continuum nonlinear elastodynamics as the interatomic distances tend to zero. Here, the continuum energy density is given by the CauchyBorn rule. The models considered allow for general finite range interactions. To control the stability of large deformations we also prove a new atomistic Gårding inequality.

Existence and convergence of solutions of the boundary value problem in atomistic and continuum nonlinear elasticity theory

“…In view of limitations and possible extensions of our work, a natural question is if analogous results can be obtained also in the dynamic setting. Based on our analysis of the static case, this problem will be addressed by the first author in the forthcoming paper [Bra16a], cf. also [Bra16b].…”

On the Stability of Objective Structures

“…An application of this atomistic stability condition is that under its assumption, solutions of the equations of continuum elasticity with smooth body forces are asymptotically approximated by the corresponding atomistic equilibrium configurations. For both the static and the dynamic case, this has been proven for small displacements on a flat torus [24,23], for the full space problem with a far-field condition [48], and for prescribed boundary values [17,15].…”

Front Matter

“…More specifically, the justification of three-dimensional elasticity from atomistic theory has been thoroughly studied by Blanc, Le Bris & Lions [2002, Alicandro & Cicalese [2004], Schmidt [2009] (for linearized elasticity), Braun & Schmidt [2013, and Braun [2017]. and the compatibility condition that there exists a vector field ϕ : Ω → E 3 such that ϕ(x) = ϕ 0 (x), x ∈ Γ 0 , and F (x) = ∇ϕ(x), x ∈ Ω.…”