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

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 \(\nabla \dot{u}\) 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\(\nabla \dot{u}\) and skew\(\nabla \dot{u}\) terms) to the reduced micro-inertia (only sym\(\nabla \dot{u}\)) 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.