Nonlinear dynamics and band gap evolution of thin-walled metamaterial-like structures

Abstract

This paper explores the dynamic behavior of metamaterial-like structures by investigating the evolution of their band gap under the influence of geometrical nonlinearities in the large displacement/rotations field. The study employs a unified framework based on the Carrera Unified Formulation (CUF) and a total Lagrangian approach to develop higher-order one-dimensional beam theories that account for geometric nonlinearities. The axis discretization is achieved through a finite element approximation. The equations of motion are solved around nonlinear static equilibrium states, which are determined using a Newton–Raphson algorithm combined with a path-following method of arc-length type. The CUF approach introduces two key innovations that are highly suitable for the evolution of the band gap:(1) Thin-walled structures can be effectively represented using a single one-dimensional beam model, overcoming the common limitations of standard finite elements. This is crucial as three-dimensional solid elements would result in significant computational costs, and two-dimensional elements based on von Kármán theory pose limitations for this type of investigation. Moreover, employing one-dimensional finite elements usually requires a combination of elements, leading to additional mathematical complexities in their connections and lacking geometric precision.(2) CUF enables the use of the full Green–Lagrange strain tensor without the need for assumptions, as is the case with von Kármán nonlinearities. The paper specifically compares results obtained with linear and nonlinear stiffness matrices, highlighting the differences. Numerical investigations are conducted on thin-walled structures composed of repeatable cells, assessing mode changes under traction and compression loading. Additionally, the paper presents a more complex metamaterial-like structure. The findings emphasize that the band gap is an inherent property of the equilibrium state, underscoring the necessity of a proper nonlinear analysis for accurately evaluating frequency transitions.