Design of two-dimensional extremal material based on truss lattices

Extremal materials with rank-deficient elastic tensor are classified by counting the soft deformation modes, while their elasticity is assigned by hard modes. To date, only pentamode materials with a single hard mode have been studied in depth, and a broader space of extremal materials is left almost unexplored. In this work, basic ingredients and design schemes aiming to more general extremal materials are studied based on truss lattices. To homogenize periodic lattices with nonaffine deformation induced by local mechanisms, we revisited the matrix formulation with singular value decomposition and derived the effective elastic tensor in compact form of Kelvin-like decomposition, with which we are able to outline the origin and design logic of extremal materials. The method is then applied to design two-dimensional extremal materials on demand of arbitrarily prescribed elasticity tensor. The unique static responses in two-dimensional unimode continua which is closely related to complex analytic functions are confirmed by our designed lattice material. The proposed method is also applicable for more general extremal materials in three-dimensional cases. [Figure not available: see fulltext.].