Modeling metamaterials by second-order rate-type constitutive relations between only the macroscopic stress and strain
Vít Průša, Casey Rodriguez, Ladislav Trnka, Martin Vejvoda
TL;DR
The paper addresses how metamaterials exhibit dynamic responses that are difficult to capture with classical, rate-independent elasticity or with naive frequency-dependent parameters. It develops a thermodynamically grounded, fully nonlinear, second-order in time rate-type constitutive framework relating the Cauchy stress $\mathbb{T}$ and the Hencky strain $\mathbb{H}$ in Eulerian coordinates, using potentials $\psi_1$ and $g_2$ to model outer and inner material components. The linearized form of the model reproduces the same dispersion relations as frequency-dependent density/stiffness theories while maintaining constant material properties in the physical domain, and a comprehensive 3D nonlinear continuum theory is derived, providing a self-contained description of metamaterial elasticity without microstructural variables. The approach offers a unified, thermodynamically consistent route to nonlinear metamaterial phenomena, with potential numerical advantages and connections to micromorphic frameworks, and sets the stage for exploring nonlinear wave and thermomechanical effects in metamaterials. The work also highlights how entropy production can be kept vanishing in mechanical processes, yielding a true elastic solid interpretation of the rate-type constitutive relations.
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.