The variational formulation of a class of elastic metamaterials with local rotational effects

This paper presents a new variational formulation for the dynamic behaviour of a class of elastic metamaterials with local rotational degrees of freedom. Unlike classical Cauchy continua, this new class of metamaterials incorporates independent local rotations as intrinsic kinematic variables, generated directly from the characteristic unit cells. This type of local rotation behaves differently from the microrotation described in the traditional micropolar model and has dramatic effects on the effective modulus of the metamaterial. In this paper, a newly unified energy functional is established in which both translational and rotational degrees of freedom contribute to free energy and kinetic energy. By applying the principle of work and energy, the complete set of governing equations is systematically derived in a rigorous and compact form. The variational formulation provides a theoretical framework capable of regenerating the metamaterial models developed in numerous previous publications that account for local rotational effects. The proposed variational structure clarifies the energetic conjugacy of the new constitutive relations containing the coupling between traditional strains and local rotations. The framework establishes a foundation for the systematic analysis and design of elastic metamaterials with local rotational effects, which could deliver the desired band gaps, wave dispersion, and other size-dependent mechanical responses.