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The effective energy of a lattice metamaterial

Xuenan Li, Robert V. Kohn

TL;DR

The paper develops a general discrete-to-continuum framework for periodic lattice metamaterials and proves that their homogenized behavior is governed by a nonlinear elastic energy $E_{\text{eff}}(u)=\int_\Omega \overline{W}(\nabla u)\,dx$ obtained via $\Gamma$-convergence. It introduces a piecewise-linear interpolation approach and an orientation-penalization term to avoid degeneracy, enabling a rigorous analysis of soft modes as deformations for which $\overline{W}(\nabla u)=0$. The main results include a comprehensive $\Gamma$-convergence theorem (with and without Dirichlet boundary data) and a variational characterization of $\overline{W}$, together with detailed applications to 2D spring lattices such as Kagome and Rotating Squares, as well as the square lattice with and without long-range springs. The work provides a general, geometry-free framework for understanding soft modes in mechanism-based metamaterials and paves the way for explicit soft-mode descriptions and design principles in future studies.

Abstract

We study the sense in which the continuum limit of a broad class of discrete materials with periodic structures can be viewed as a nonlinear elastic material. While we are not the first to consider this question, our treatment is more general and more physical than those in the literature. Indeed, it applies to a broad class of systems, including ones that possess mechanisms; and we discuss how the degeneracy that plagues prior work in this area can be avoided by penalizing change of orientation. A key motivation for this work is its relevance to mechanism-based mechanical metamaterials. Such systems often have ``soft modes'', achieved in typical examples by modulating mechanisms. Our results permit the following more general definition of a soft mode: it is a macroscopic deformation whose effective energy vanishes -- in other words, one whose spatially-averaged elastic energy tends to zero in the continuum limit.

The effective energy of a lattice metamaterial

TL;DR

The paper develops a general discrete-to-continuum framework for periodic lattice metamaterials and proves that their homogenized behavior is governed by a nonlinear elastic energy obtained via -convergence. It introduces a piecewise-linear interpolation approach and an orientation-penalization term to avoid degeneracy, enabling a rigorous analysis of soft modes as deformations for which . The main results include a comprehensive -convergence theorem (with and without Dirichlet boundary data) and a variational characterization of , together with detailed applications to 2D spring lattices such as Kagome and Rotating Squares, as well as the square lattice with and without long-range springs. The work provides a general, geometry-free framework for understanding soft modes in mechanism-based metamaterials and paves the way for explicit soft-mode descriptions and design principles in future studies.

Abstract

We study the sense in which the continuum limit of a broad class of discrete materials with periodic structures can be viewed as a nonlinear elastic material. While we are not the first to consider this question, our treatment is more general and more physical than those in the literature. Indeed, it applies to a broad class of systems, including ones that possess mechanisms; and we discuss how the degeneracy that plagues prior work in this area can be avoided by penalizing change of orientation. A key motivation for this work is its relevance to mechanism-based mechanical metamaterials. Such systems often have ``soft modes'', achieved in typical examples by modulating mechanisms. Our results permit the following more general definition of a soft mode: it is a macroscopic deformation whose effective energy vanishes -- in other words, one whose spatially-averaged elastic energy tends to zero in the continuum limit.
Paper Structure (27 sections, 12 theorems, 294 equations, 9 figures)

This paper contains 27 sections, 12 theorems, 294 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Mechanism-based mechanical metamaterials
  3. Overview of our framework and results
  4. Getting started
  5. Lattice nodes, some basic examples, and Lg
  6. The piecewise linearization of a deformation, our basic energy bounds, and Lg
  7. The scaled lattice, Lg, and our admissible deformations
  8. Boundary conditions, and gluing deformations together
  9. Statements of our main results
  10. The proof of the main theorem
  11. The heart of the matter: affine limits
  12. Some easy properties of Lg
  13. The proof of Theorem \ref{['thm:main-theorem']}
  14. The proof of Theorem \ref{['thm:theorem-dirichlet']}
  15. Applications to 2D lattice systems of springs
  16. ...and 12 more sections

Key Result

Lemma 1

Suppose a collection of scaled unit cells $\{\epsilon U + \alpha^{(j)}\}_{j=1}^P$, a deformation $u^\epsilon$, a domain $\Omega$, and a constant $M$ have the properties that Then

Figures (9)

  • Figure 1: One of the periodic mechanisms on the Kagome lattice: (a) the reference state; (b) the deformed state.
  • Figure 2: Soft modes of the Kagome lattice: (a) the reference state; (b) a modulated version of the mechanism shown in \ref{['fig:kagome-one-periodic']}; (c) a modulated version of a different mechanism. The color in each plot indicates the rotation angle of local equilateral triangles.
  • Figure 3: (a) The Kagome lattice: The shaded rectangle represents the unit cell $U$ for the Kagome lattice, which contains three vertices $A,O,D$ marked in red. These vertices can be translated to obtain the entire lattice. The solid red edges are those included in the energy calculation $E(u,U)$ in equation \ref{['eqn:kagome-intro-energy']}. A translated copy of these edges is marked in yellow to illustrate that all edges in the Kagome lattice can be viewed as translated copies of the red solid edges. The dotted lines indicate the triangular mesh used to interpolate the admissible deformations. (b) The square lattice with long-range interactions: the same coloring scheme is used to describe the unit cell, vertices within the unit cell, edges contributing to the energy $E(u,U)$, and the triangular mesh. The details are similar to those described for the Kagome lattice and are omitted for brevity.
  • Figure 4: A polygon with $n=7$: the energy on the red solid edges are counted in $E_{\text{poly}}(u, P_n)$, while the dotted edges are artificial edges to indicate the triangular mesh. Here the vertices are numbered counter-clockwise, however our upper and lower bounds are also valid (with the same proofs) when the vertices are numbered clockwise.
  • Figure 5: The unit cells of the Kagome lattice and the Rotating Squares lattice: nodes in $\mathcal{V}$ are marked in red; nodes not in $\mathcal{V}$ but used in the energy $E(u,U)$ are marked in cyan; edges whose springs are counted in $E(u,U)$ are marked in red solid lines; artificial edges used to define the triangular mesh are marked in cyan dotted lines; the shaded area is $U$.
  • ...and 4 more figures

Theorems & Definitions (32)

  • Remark 1
  • Definition 1
  • Lemma 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 2
  • Remark 2
  • Remark 3
  • ...and 22 more