A new continuum model of a class of elastic metamaterials with local rotational effects

Elastic metamaterials are typically synthesized in the form of a series of periodic unit cells, which are smaller than the typical phenomenological length scale. The development of continuum models capable of capturing the size effect of the unit cells has proven to be notoriously difficult. This paper focuses on the development of a novel 2-D continuum model capable of capturing the dynamic behaviour of a class of elastic metamaterials in the long wavelength limit. A new constitutive relation incorporating the local rotational effects is proposed, and a representative discrete model using linear Hookean springs and rigid disks is introduced to generate a continuum metamaterial model revealing the newly proposed constitutive relation. The new model is compared with classical models such as the micropolar continuum model, and other more recently proposed 1-D models for these types of elastic metamaterials. General 2-D wave propagation is then explored analytically using the newly developed model through analysis of the dispersion relations for both longitudinal and transverse waves.