A matrix method for designing bandgap of stretch-dominated mechanical metamaterials

The dynamic equation in matrix form is established for the unit cell of stretch-dominated mechanical metamaterials, which is modelled as a pin-bar structure. By condensing the degrees of freedom of equivalent nodes by introducing Bloch’s theorem, the dynamic equation is transformed into a generalized characteristic equation for solving the dispersion function of metamaterials. It is proven that the necessary condition for a bandgap to exist is that the number of inequivalent nodes of the metamaterial is greater than 1 by analysing the generalized characteristic equation. The formula for calculating the number of inequivalent nodes is given, which indicates that it can generally be increased either by inserting nodes or by redefining the unit cell. Based on matrix perturbations and the gradient descent method, a numerical strategy is proposed to generate a bandgap with an expected width or range by optimizing the element cross-sectional areas and node coordinates of the unit cell. The degenerate points of the dispersion function can also be eliminated by using matrix perturbations. Two metamaterials are employed as examples to verify the validity of the proposed method. The prescribed bandgaps are generated by increasing the number of inequivalent nodes and optimizing the element cross-sectional areas and node coordinates.