Retrieval method for elastic Willis metamaterials with mode coupling

In this paper, a method for elastic Willis metamaterials that can retrieve effective parameters even when there is mode coupling is proposed. While recent progress has enlarged the research scope of acoustic and elastic metamaterials to Willis metamaterials, in which stresses and velocities have unique couplings, research on elastic Willis metamaterials has been relatively rare due to its complicated physics. Elastic metamaterials, especially, have more parameters due to the existence of the shear wave mode and the mode coupling between longitudinal and shear wave modes. If elastic Willis metamaterials have mode coupling, the governing equations become too complicated to be analyzed with the existing parameter retrieval method. Here, we propose a method that avoids the complicated equations and effectively retrieves the parameters. Our procedure is based on a transfer matrix decomposition involving eigenvalues and eigenvectors. As a validation, both a comparison to an ideal mass-spring system and the application to continuum elastic Willis metamaterials are performed. With the proposed method, we believe that new research areas based on elastic Willis metamaterials could be possible.