Complete characterization of the macroscopic deformations of periodic unimode metamaterials of rigid bars and pivots

TL;DR

This paper provides a comprehensive analysis of the macroscopic deformation capabilities of periodic unimode metamaterials made from rigid bars and pivots, demonstrating their potential for large, auxetic deformations with zero bulk modulus.

Contribution

It introduces a complete characterization of possible deformations in unimode metamaterials and constructs examples of highly flexible, auxetic dilational materials with large strain capabilities.

Findings

Any continuous deformation trajectory can be approximated without collapse.

Two and three dimensional dilational materials exhibit large flexibility windows.

These materials are perfect auxetics with zero bulk modulus.

Abstract

A complete characterization is given of the possible macroscopic deformations of periodic nonlinear affine unimode metamaterials constructed from rigid bars and pivots. The materials are affine in the sense that their macroscopic deformations can only be affine deformations: on a local level the deformation may vary from cell to cell. Unimode means that macroscopically the material can only deform along a one dimensional trajectory in the six dimensional space of invariants describing the deformation (excluding translations and rotations). We show by explicit construction that any continuous trajectory is realizable to an arbitrarily high degree of approximation provided at all points along the trajectory the geometry does not collapse to a lower dimensional one. In particular, we present two and three dimensional dilational materials having an arbitrarily large flexibility window. These are perfect auxetic materials for which a dilation is the only easy mode of deformation. They are free to dilate to arbitrarily large strain with zero bulk modulus.