A two-step strategy for designing stretch-dominated metamaterials simultaneously satisfying desired elastic tensor and bandgap

In engineering, the design of bandgap mechanical metamaterials is generally required to satisfy the necessary static properties, which are macroscopically characterized by an elastic tensor. In this work, a design strategy is developed for stretch-dominated mechanical metamaterials that need to simultaneously satisfy the desired elastic tensors and bandgaps by modelling them as pin-jointed bar structures. Considering that the elastic tensor of a metamaterial is determined by the stiffness matrix of its unit cell, an analytical relationship between them is established using the Cauchy‒Born rule. Based on the analysis of the stiffness matrix, a sensitivity expression of the elastic tensor with respect to the element cross-sectional areas and the node coordinates of the unit cell is established, which can be used to design two types of unit cell parameters for the desired elastic tensor. This sensitivity matrix has a broad null space because the number of unit cell parameters to be designed is generally much larger than that of the desired elastic tensor components. Using the basis vectors of this null space, these design parameters can be modified to create the desired bandgap while keeping the elastic tensor components constant. Even if the elastic tensor components deviate during numerical iteration, they can be corrected by adjusting the design parameters in the null space of the bandgap sensitivity matrix. Thus, a two-step strategy is proposed to design the elastic tensor and the bandgap sequentially. The proposed method is verified by an example in which the desired elastic tensor components of a three-dimensional metamaterial are designed and two examples in which both the desired elastic tensor components and bandgaps are achieved. The elastic tensors of the obtained metamaterials are verified by comparing the deformations of the metamaterials and those of their equivalent continuum finite element models, and the bandgaps are also confirmed by the frequency–response curves.