Null-lagrangians in the micro-inertia contribution of the reduced relaxed micromorphic model. Theoretical and computational insights with applications

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This paper identifies a null-Lagrangian in the reduced relaxed micromorphic model. We show that the introduction of a micro-inertia depending on the skew-symmetric part of ∇u˙ with the macroscopic displacement field u does not enrich the dispersion relations of the reduced relaxed micromorphic model. Reciprocally, we show that one can switch from the full micro-inertia (with both sym∇u˙ and skew∇u˙ terms) to the reduced micro-inertia (only sym∇u˙) without any additional fitting. This is related to the fact that the introduction of such a skew-symmetric term is equivalent to a null-Lagrangian that leaves the bulk response unchanged while modifying the Neumann boundary conditions at the boundaries. Thus, the introduction of the skew-symmetric part of the micro-inertia, while redundant for wave dispersion, may potentially be used to improve the response of finite-size mechanical metamaterials at the homogenized macroscale due to boundary effects.