Design of smart metamaterials for vibration control: extension of Bloch approach to handle finite system boundary conditions

The design of smart metamaterials for vibration control is usually based on the use of Bloch theorem which considers a single cell with adequate boundary conditions. These boundary conditions correspond to the in finite repetition of the unit cell in 1D, 2D or 3D. Complex geometries and composite systems can then be designed using this approach with finite elements. The control of the elastic waves can be performed by combining Bragg's (wave interferences), resonant's (resonance of a component embedded in the unit cell), damping and/or active control. The energy can then be reflected, transmitted, damped, focused or con fined in a specific zone of the structure. However, the practical realization of real-life 2D or 3D finite systems may lead to some situations where energy transfers are not in accordance with those predicted by the in finite system considered in the design, because of reflections on the boundary conditions of the finite structure. The behavior of the system may be simulated by full system modelling, but this is time consuming and may lead to huge calculation costs. In this paper, we propose an extension of the Bloch approach to handle finite system boundary conditions in order to be able to identify situations in which energy transfer may arise because of reflections on the border of the elastic domain. Calculations are performed on 2 cells with adequate boundary conditions. The methodology is described and validated using full finite model and experimental tests on a 2D metamaterial structure.