Modeling metamaterials by second-order rate-type constitutive relations between only the macroscopic stress and strain

TL;DR

This paper introduces a new thermodynamically consistent second-order rate-type constitutive model for metamaterials that captures finite deformations without microstructural assumptions, aligning with frequency-dependent behaviors using constant material properties.

Contribution

It presents a novel second-order in time rate-type constitutive relation for metamaterials that does not rely on microstructural concepts or frequency-dependent parameters.

Findings

Reproduces frequency-dependent behavior with constant properties.

Does not require microstructural or enriched continuum concepts.

Defines a new class of elastic solid models.

Abstract

We propose a thermodynamically based approach for constructing effective rate-type constitutive relations describing finite deformations of metamaterials. The effective constitutive relations are formulated as \emph{second-order} in time rate-type Eulerian constitutive relations between only the Cauchy stress tensor, the Hencky strain tensor and objective time derivatives thereof. In particular, there is no need to introduce additional quantities or concepts such as ``micro-level deformation'',``micromorphic continua'', ``enriched continua'', or elastic solids with frequency dependent material properties. The linearisation of the proposed fully nonlinear (finite deformations) constitutive relations leads, in Fourier space, to the same constitutive relations as those commonly used in theories based on the concepts of frequency dependent density and/or stiffness. From this perspective the proposed constitutive relations reproduce the behaviour predicted by the frequency dependent density and/or stiffness models, but yet they work with constant -- that is motion independent -- material properties. Finally, we argue that the proposed fully nonlinear (finite deformations) second-order in time rate-type constitutive relations do not fall into traditional classes of models for elastic solids (hyperelastic solids/Green elastic solids, first-order in time hypoelastic solids), and that the proposed constitutive relations embody a \emph{new} class of constitutive relations characterising elastic solids.