Wave propagation in auxetic mechanical metamaterial: Bloch formalism for various boundary conditions

Abstract

The auxetic meta-material is a special class of macro-structure designed for exhibiting negative Poisson's ratio. The spatial repetition of the lattice affects the wave propagation and dynamic responses. In this study, a mathematical basis of Bloch analysis for auxetic media have been presented and shaded some important light on the application of the technique. For this design an analysis has been carried out to show how the periodic boundary condition changes with the connectivity, orientation and geometry of unit lattice. The corresponding eigen value problem is developed to obtain the propagation frequency which is the function of mass, stiffness matrix and the wave vector. Mapping of different forms which are associated with the nodal displacements of a unit cell to adjacent cell have been demonstrated and formulated mathematically and observed its effect on the reduced stiffness and mass matrices is studied. Modal analysis has been carried out using Abaqus 6.14 and the transverse nodal displacements obtained from the F.E analysis. The Bloch formulation for revolving-square type auxetic structure has been formulated and validated. Further, we have obtained the changes in auxetic behavior of the structure under different boundary conditions. The periodicity of a given lattice assists in determining the frequency bands within which the propagation of elastic waves is permitted. Further study is proposed for attenuation and transmission band analysis to steer the waves which roots the idea of vibration sink, in which we can damp the mechanical waves effectively at a specific location. The Auxetic meta-materials is envisaged to have a significant role in wave attenuation and wave steering, and can be used effectively for vibration control. © 2019 Copyright SPIE.