Abstract： Metamaterials with zero-energy mode are a kind of elastic material that some eigenvalues of the elastic matrix are zero. By counting the number of zero eigenvalues, they are classified as from uni-mode to penta-mode material. To date, only penta-mode materials have been studied in depth and found important applications in manipulation of underwater acoustic wave and elastic wave, while other types of material with zero-energy modes remain almost untouched. In this study, we present a comprehensive development for design of two-dimensional uni-mode material based on periodic distorted Kagome truss lattices. By using the Cauchy-Born hypothesis and matrix formulation of truss systems, we developed a homogenization method for general lattices which are under-constrained. Under the macroscopic strain field, the method can take care of non-affine relaxation due to the microscopic mechanism, thus can correctly predict the rank deficient effective elastic tensor. Further, the relation between the microscopic self-stress and mechanism states as well as the macroscopic hard and soft modes is clarified. In particular, for distorted Kagome lattices, we are able to analytically express the soft mode by irreducible geometric parameters, while the rest parameters including the bar stiffnesses are responsible only for hard modes. To match a given elastic tensor, we proposed a two-level design scheme to seek microstructural parameters with high efficiency and accuracy. It is revealed that the distorted Kagome lattice is able to realize a wide spectrum of uni-mode materials via tuning its configuration. Finally, the developed method was verified in conjunction with unusual wave behaviors found in uni-mode materials, and excellent agreement was achieved between the theoretical and numerical predictions. We found that the slowness curve of an uni-mode material may possess opened shape, which is not found in ordinary orthotropic materials, and can be utilized in wideband negative refraction of elastic wave beam. The work may initiate design of more general metamaterials with zero-energy mode, and may inspire further explorations on applications other than those of penta-mode material.
2022, Vol. 43 
