Size effects in nonlinear periodic materials exhibiting reversible pattern transformations

TL;DR

This study investigates how size influences the behavior of nonlinear periodic metamaterials undergoing reversible pattern transformations, revealing limitations of classical homogenization at small scales.

Contribution

It introduces a comprehensive analysis of size effects in nonlinear metamaterials with reversible pattern transformations, considering microstructure arrangements and boundary conditions.

Findings

Classical homogenization matches ensemble solutions only at large scale ratios.

Size effects become significant at small scale ratios, invalidating classical homogenization.

Boundary conditions influence pattern transformation and size effects.

Abstract

This paper focuses on size effects in periodic mechanical metamaterials driven by reversible pattern transformations due to local elastic buckling instabilities in their microstructure. Two distinct loading cases are studied: compression and bending, in which the material exhibits pattern transformation in the whole structure or only partially. The ratio between the height of the specimen and the size of a unit cell is defined as the scale ratio. A family of shifted microstructures, corresponding to all possible arrangements of the microstructure relative to the external boundary, is considered in order to determine the ensemble averaged solution computed for each scale ratio. In the compression case, the top and the bottom edges of the specimens are fully constrained, which introduces boundary layers with restricted pattern transformation. In the bending case, the top and bottom edges are free boundaries resulting in compliant boundary layers, whereas additional size effects emerge from imposed strain gradient. For comparison, the classical homogenization solution is computed and shown to match well with the ensemble averaged numerical solution only for very large scale ratios. For smaller scale ratios, where a size effect dominates, the classical homogenization no longer applies.