On the emergence of negative effective density and modulus in 2-phase phononic crystals

Abstract

In this paper we report metamaterial properties including negative and singular effective properties for what would traditionally be considered non locally resonant 2-phase phononic unit cells. The negative effective material properties reported here occur well below the homogenization limit and are, therefore, acceptable descriptions of overall behavior. The material property combinations which make this possible were first revealed by a novel level set based topology optimization process which we describe. The optimization process revealed that a 2-phase unit cell in which one of the phases is simultaneously lighter and stiffer than the other results in dynamic behavior which has all the attendant characteristics of a locally resonant composite including negative effective properties far below the homogenization limit. We investigate this further using the Craig–Bampton decomposition and clarify that these properties emerge through an interplay between the fundamental internal modeshape of the unit cell and a rigid body mode. Through explicit numerical calculations on 1-D, 2-phase unit cells, we show that negative effective properties only appear for the specific material property combination mentioned above. Furthermore, we provide a proof which supports this conclusion. The concept is also shown to hold for 2-D unit cells where we show that an appropriately designed hexagonal unit cell made of 2 material phases exhibits negative effective shear modulus and density in an appropriate frequency regime in which it also exhibits negative refraction. As in the 1-D case, we show that the homogenized description in the 2-D case is rigorous as well. An important conclusion of this paper is that the class of unit cells expected to result in negative properties can be expanded beyond the classic unit cell (three-phase unit cells with an explicit locally resonant phase) to include topologically simpler 2-phase unit cells as well.