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A nonlinear homogenization-based perspective on the soft modes and effective energies of some conformal metamaterials

Xuenan Li, Robert V. Kohn

TL;DR

This work uses nonlinear homogenization to study mechanism-based 2D metamaterials, focusing on Kagome and Rotating Squares lattices. By formulating a lattice energy with an orientation penalty and proving a Gamma-convergence to an effective energy density $\overline{W}^\eta$, it identifies soft modes as deformations with vanishing $\overline{W}^\eta$; for small penalty $0<\eta<\eta_0$, the zero set corresponds exactly to isotropic compressions, i.e., deformations with $Du=cR$ for $0\le c\le 1$ and $R\in SO(2)$, i.e., compressive conformal maps. The authors establish a robust two-part lower bound on $\overline{W}^\eta(\lambda)$ capturing both isotropy and compressivity, and show these results extend to other conformal metamaterials. The approach leverages the structure and symmetry of periodic mechanisms to bound the effective energy from below, providing a rigorous framework for predicting soft-mode behavior and guiding design of auxetic/metamaterial devices.

Abstract

There is a growing mechanics literature concerning the macroscopic properties of mechanism-based mechanical metamaterials. This amounts mathematically to a homogenization problem involving nonlinear elasticity. A key goal is to identify the "soft modes" of the metamaterial. We achieve this goal using methods from homogenization for some specific 2D examples -- including discrete models of the Rotating Squares metamaterial and the Kagome metamaterial -- whose soft modes are compressive conformal maps. The innovation behind this achievement is a new technique for bounding the effective energy from below, which takes advantage of the metamaterial's structure and symmetry.

A nonlinear homogenization-based perspective on the soft modes and effective energies of some conformal metamaterials

TL;DR

This work uses nonlinear homogenization to study mechanism-based 2D metamaterials, focusing on Kagome and Rotating Squares lattices. By formulating a lattice energy with an orientation penalty and proving a Gamma-convergence to an effective energy density , it identifies soft modes as deformations with vanishing ; for small penalty , the zero set corresponds exactly to isotropic compressions, i.e., deformations with for and , i.e., compressive conformal maps. The authors establish a robust two-part lower bound on capturing both isotropy and compressivity, and show these results extend to other conformal metamaterials. The approach leverages the structure and symmetry of periodic mechanisms to bound the effective energy from below, providing a rigorous framework for predicting soft-mode behavior and guiding design of auxetic/metamaterial devices.

Abstract

There is a growing mechanics literature concerning the macroscopic properties of mechanism-based mechanical metamaterials. This amounts mathematically to a homogenization problem involving nonlinear elasticity. A key goal is to identify the "soft modes" of the metamaterial. We achieve this goal using methods from homogenization for some specific 2D examples -- including discrete models of the Rotating Squares metamaterial and the Kagome metamaterial -- whose soft modes are compressive conformal maps. The innovation behind this achievement is a new technique for bounding the effective energy from below, which takes advantage of the metamaterial's structure and symmetry.

Paper Structure

This paper contains 17 sections, 14 theorems, 148 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Problem setup and main results
  3. The Rotating Squares and Kagome metamaterials
  4. Introduction to some lattice systems of springs
  5. The scaled energy and our admissible deformations
  6. Existence of an effective energy
  7. Our main results
  8. A geometric argument: periodic mechanisms are isotropic compressions
  9. Deformations of the form $u(x) = \lambda x + \psi$ with $k$-periodic $\psi$
  10. The proof of Proposition \ref{['lemma-geometric']}
  11. Lower bounds for the effective energy density $\overline{W}^\eta(\lambda)$
  12. The averaged energy on $kU$
  13. The isotropic bound
  14. The compressive bound
  15. Other conformal metamaterials
  16. ...and 2 more sections

Key Result

Theorem 2.4

Let $E^{\epsilon,\eta} (u^\epsilon, \Omega)$ be our discrete energy for either the Kagome or the Rotating Squares metamaterial, for any $\eta > 0$. Then for any bounded, Lipschitz domain $\Omega$, $E^{\epsilon,\eta}(u,\Omega)$$\Gamma$-converges in $H^1(\Omega)$ as $\epsilon \rightarrow 0$ (with resp Moreover, the effective energy density $\overline{W}^\eta (\lambda)$ is independent of the domain $

Figures (18)

  • Figure 1: The Rotating Squares metamaterial: (a) its reference state as a cut-out and its image under the mechanism; (b) its reference state as a spring network, with springs connecting the diagonals of certain squares.
  • Figure 2: The reference state of the Kagome metamaterial, viewed (a) as a cut-out, and (b) as a lattice of springs in which each node is connected to its nearest neighbors by springs.
  • Figure 3: Some mechanisms of the Kagome metamaterial: (a) the deformed state of the one-periodic mechanism $u_\theta$, controlled by a single angle $\theta$; (b) the deformed state of a two-periodic mechanism $u_{\theta_1, \theta_2, \theta_3}$, controlled by three angles $\theta_1, \theta_2, \theta_3$ (for more detail see li2023some); (c) the deformed state of a non-periodic mechanism $u_\theta$, controlled by a single angle $\theta$ (for more detail see Section \ref{['subsec:non-periodic']}).
  • Figure 4: Soft modes in the Kagome lattice: (a) a rectangle filled with the Kagome metamaterial in its reference state; (b) the image of a soft mode, achieved microscopically by modulating the mechanism shown in Figure \ref{['fig:kagome-mechanisms']}a; (c) the image of the same soft mode, achieved microscopically by modulating a different mechanism. The colors in (b) and (c) indicate the rotation angles of the deformed structure's triangles.
  • Figure 5: (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 $E_\text{spr}(u,U)$ in equation \ref{['eqn:kagome-spr-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 Rotating Squares lattice. The nodes associated with the unit cell (our set $V$) are marked in red; nodes not in $V$ but used in the energy $E(u,U)$ are marked in cyan; springs counted in $E_\text{spr}(u,U)$ are marked by red solid lines; artificial edges used only for the triangularization of $U$ are marked by cyan dotted lines; the shaded area is $U$.
  • ...and 13 more figures

Theorems & Definitions (34)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3: $\Gamma$-convergence
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7: Compressive conformal maps
  • Remark 2.8
  • Theorem 2.9
  • ...and 24 more