| Abstract: |
| We discuss a novel thermodynamically based approach for constructing effective rate-type constitutive relations describing finite deformations of \emph{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 dispersion relations as those commonly used in metamaterial 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. The behaviour predicted by the linearised models is documented through numerical experiments exploiting a recently proposed dispersion-relation preserving discretisation.
Finally, we argue that the proposed fully nonlinear (finite deformations) second-order in time rate-type constitutive relations do not fall into the 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 new class of constitutive relations characterising elastic solids. |
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