Fracture and size effect in mechanical metamaterials

TL;DR

This paper develops a variational framework to analyze how the size of cells in mechanical metamaterials influences their fracture behavior, revealing size-dependent toughness and a lattice shielding mechanism.

Contribution

It introduces a higher-order asymptotic analysis using b3-expansion to explicitly quantify size effects and fracture mechanics in metamaterials.

Findings

Metamaterials behave as micropolar continua with no size effect at zeroth order.

Second-order corrections reveal size-dependent toughness and lattice shielding.

The theory aligns with experimental observations of anti-shielding effects.

Abstract

We resort to variational methods to evaluate the asymptotic behavior of fine metamaterials as a function of cell size. To zeroth order, the metamaterial behaves as a micropolar continuum with both displacement and rotation degrees of freedom, but exhibits linear-elastic fracture mechanics scaling and therefore no size effect. To higher order, the overall energetics of the metastructure can be characterized explicitly in terms of the solution of the zeroth-order continuum problem by the method of {\Gamma}-expansion. We present explicit expressions of the second-order correction for octet frames. As an application, we evaluate the compliance of double-cantilever octet specimens to second order and use the result to elucidate the dependence of the apparent toughness of the specimen on cell size. The analysis predicts the discreteness of the metamaterial lattice to effectively shield the crack-tip, a mechanism that we term lattice shielding. The theory specifically predicts anti-shielding, i. e., coarser is weaker, in agreement with recent experimental observations.