基于桁架格栅的二维模式材料设计方法

Design of two-dimensional extremal material based on truss lattices

具有不满秩弹性张量的模式材料(extremal material)可以根据其具有的软模式(soft deformation mode或者soft mode)数目进行分类,其具体的弹性性质则由其硬模式(hard modes)决定.迄今为止,只有具有单一硬模式的五模材料得到了深入的研究,包含其他模式材料的更广阔空间则几乎未被探索.本文基于桁架格栅,研究了设计更一般的模式材料的基本工具和设计方案.为了均匀化具有由局部机构引起的非仿射变形的周期格栅,我们回顾了奇异值分解的矩阵公式,并推导了形如开尔文分解的、紧凑形式的有效弹性张量,由此我们能够概述模式材料的产生机理和设计逻辑.我们将该方法应用于具有任意指定的弹性张量的二维模式材料的设计.所设计的格栅材料证实了理想二维一模材料中,与解析复变函数密切相关的、独特的静态响应.本文所提出的方法也适用于三维情况下更一般的模式材料.

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.