Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrete-to-continuum linearization in atomistic dynamics

Manuel Friedrich, Manuel Seitz, Ulisse Stefanelli

Abstract

In the stationary case, atomistic interaction energies can be proved to $Γ$-converge to classical elasticity models in the simultaneous atomistic-to-continuum and linearization limit [19],[40]. The aim of this note is that of extending the convergence analysis to the dynamic setting. Moving within the framework of [40], we prove that solutions of the equation of motion driven by atomistic deformation energies converge to the solutions of the momentum equation for the corresponding continuum energy of linearized elasticity. By recasting the evolution problems in their equivalent energy-dissipation-inertia-principle form, we directly argue at the variational level of evolutionary $Γ$-convergence [32],[36]. This in particular ensures the pointwise in time convergence of the energies.

Discrete-to-continuum linearization in atomistic dynamics

Abstract

In the stationary case, atomistic interaction energies can be proved to -converge to classical elasticity models in the simultaneous atomistic-to-continuum and linearization limit [19],[40]. The aim of this note is that of extending the convergence analysis to the dynamic setting. Moving within the framework of [40], we prove that solutions of the equation of motion driven by atomistic deformation energies converge to the solutions of the momentum equation for the corresponding continuum energy of linearized elasticity. By recasting the evolution problems in their equivalent energy-dissipation-inertia-principle form, we directly argue at the variational level of evolutionary -convergence [32],[36]. This in particular ensures the pointwise in time convergence of the energies.
Paper Structure (19 sections, 13 theorems, 101 equations, 1 figure)

This paper contains 19 sections, 13 theorems, 101 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting of the problem and main result
  3. Lattice model
  4. Main result
  5. Convergence of solutions
  6. $\Gamma$-convergence and compactness
  7. Gradients of the energy and auxiliary results
  8. Fréchet differential of $I$
  9. Approximation by scaled lattice displacements
  10. A priori estimates
  11. Compactness and limit of $\nu \dot{u}_k$
  12. Limit of $\rho \ddot{u}_k + \partial I_k(u_k)$
  13. Limit of $\partial I_k (u_k)$
  14. Limit of $\rho \ddot{u}_k$
  15. Joint limit
  16. ...and 4 more sections

Key Result

Theorem 2.3

Suppose that Assumption assumptions holds and let $\varepsilon_k$ and $\delta_k$ be such that $\varepsilon_k \to 0$ and $\delta_k \to 0$. Convergence of solutions: Let $u_\varepsilon^0, u_\varepsilon^1 \in \mathcal{A}_\varepsilon$ be an initial scaled lattice displacement and velocity, and let

Figures (1)

  • Figure 1: Scaled reference configuration of the atomistic system. Here, the lattice is $_\varepsilon = \varepsilon \mathbb{Z}^2$, i.e., $A$ is the identity matrix. The labeling of the corner points of a scaled cell is indicated by the vectors $\varepsilon z_i$, $i = 1, \dots, 4$.

Theorems & Definitions (30)

  • Remark 2.2
  • Theorem 2.3: Discrete-to-continuum linearization
  • Remark 2.4: Purely viscous case
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3: Symmetry of $\mathbb{C}$
  • proof
  • Remark 3.4: $L^2$-regularity of $\partial I (w)$
  • Lemma 3.5: Approximation by scaled lattice displacements
  • proof
  • ...and 20 more