Customization of two-dimensional extremal materials

Extremal materials are elastic metamaterials with rank-deficient elasticity tensor. Due to this novel property of the elasticity tensor, extremal materials (especially pentamode materials) show outstanding performance in various wave manipulation applications. However, until now there is still a lack of a systematic and general design methodology that is applicable for all kinds of extremal materials. In this study, we proposed to design extremal materials by using topology optimization techniques without any prior knowledge about the geometric symmetry of the unit cell, thus more anisotropic materials can in principle be designed. Besides the number of zero eigenvalues of the elasticity tensor, the corresponding eigenvectors (usually called the soft modes of the extremal materials) can also be customized in our method. We use a two-step design strategy to accomplish the design of the microstructures. Several numerical examples are shown to verify the effectiveness of the optimization framework in designing extremal materials with and without constraints on soft/hard modes. Numerical results reveal that the proposed method can design extremal materials with arbitrary values of soft or hard modes. The proposed methodology can be easily extended to three-dimensional case. Our works pave the way for exploring the rich property of extremal materials.